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In [[abstract algebra]], the '''focal subgroup theorem''' describes the fusion of elements in a [[Sylow subgroup]] of a [[finite group]].  The focal subgroup theorem was introduced in {{harv|Higman|1958}} and is the "first major application of the transfer" according to {{harv|Gorenstein|Lyons|Solomon|1996|p=90}}. The focal subgroup theorem relates the ideas of transfer and fusion such as described in {{harv|Grün|1935}}. Various applications of these ideas include local criteria for [[p-nilpotent group|''p''-nilpotence]] and various non-simplicity criterion focussing on showing that a finite group has a [[normal subgroup]] of index ''p''.
 
== Background ==
 
The focal subgroup theorem relates several lines of investigation in finite group theory: normal subgroups of index a power of ''p'', the transfer homomorphism, and fusion of elements.
 
=== Subgroups ===
The following three normal subgroups of index a power of ''p'' are naturally defined, and arise as the smallest normal subgroups such that the quotient is (a certain kind of) ''p''-group. Formally, they are kernels of the reflection onto the [[reflective subcategory]] of ''p''-groups (respectively, elementary abelian ''p''-groups, abelian ''p''-groups).
* '''E'''<sup>''p''</sup>(''G'') is the intersection of all index ''p'' normal subgroups; ''G''/'''E'''<sup>''p''</sup>(''G'') is an elementary abelian group, and is the largest elementary abelian ''p''-group onto which ''G'' surjects.
* '''A'''<sup>''p''</sup>(''G'') (notation from {{harv|Isaacs|2008|loc=5D, p. 164}}) is the intersection of all normal subgroups ''K'' such that ''G''/''K'' is an abelian ''p''-group (i.e., ''K'' is an index <math>p^k</math> normal subgroup that contains the derived group <math>[G,G]</math>): ''G''/'''A'''<sup>''p''</sup>(''G'') is the largest abelian ''p''-group (not necessarily elementary) onto which ''G'' surjects.
* '''O'''<sup>''p''</sup>(''G'') is the intersection of all normal subgroups ''K'' of ''G'' such that ''G''/''K'' is a (possibly non-abelian) ''p''-group (i.e., ''K'' is an index <math>p^k</math> normal subgroup): ''G''/'''O'''<sup>''p''</sup>(''G'') is the largest ''p''-group (not necessarily abelian) onto which ''G'' surjects. '''O'''<sup>''p''</sup>(''G'') is also known as the {{anchor|p-residual subgroup}}'''''p''-residual subgroup'''.
Firstly, as these are weaker conditions on the groups ''K,'' one obtains the containments <math>\mathbf{E}^p(G) \supseteq \mathbf{A}^p(G) \supseteq \mathbf{O}^p(G).</math> These are further related as:
:'''A'''<sup>''p''</sup>(''G'') = '''O'''<sup>''p''</sup>(''G'')[''G'',''G''].
'''O'''<sup>''p''</sup>(''G'') has the following alternative characterization as the subgroup generated by all Sylow ''q''-subgroups of ''G'' as ''q''≠''p'' ranges over the prime divisors of the [[order (group theory)|order]] of ''G'' distinct from ''p''.  
 
'''O'''<sup>''p''</sup>(''G'') is used to define the '''lower ''p''-series'''{{anchor|lower p-series}} of ''G'', similarly to the [[upper p-series|upper ''p''-series]] described in [[p-core]].
 
=== Transfer homomorphism ===
{{main|Transfer (group theory)}}
The '''transfer homomorphism''' is a homomorphism that can be defined from any group  ''G'' to the abelian group ''H''/[''H'',''H''] defined by a subgroup ''H'' ≤ ''G'' of [[finite index]], that is [''G'':''H''] &lt; ∞.  The transfer map from a finite group ''G'' into its Sylow ''p''-subgroup has a [[kernel (group theory)|kernel]] that is easy to describe:
:The kernel of the transfer homomorphism from a finite group ''G'' into its Sylow ''p''-subgroup ''P'' has '''A'''<sup>''p''</sup>(''G'') as its kernel, {{harv|Isaacs|2008|loc=Theorem 5.20, p. 165}}.
In other words, the "obvious" homomorphism onto an abelian ''p''-group is in fact the most general such homomorphism.
 
=== Fusion ===
The '''fusion''' pattern of a subgroup ''H'' in ''G'' is the equivalence relation on the elements of ''H'' where two elements ''h'', ''k'' of ''H'' are '''fused''' if they are ''G''-conjugate, that is, if there is some ''g'' in ''G'' such that ''h'' = ''k''<sup>''g''</sup>. The normal structure of ''G'' has an effect on the fusion pattern of its Sylow ''p''-subgroups, and conversely the fusion pattern of its Sylow ''p''-subgroups has an effect on the normal structure of ''G'', {{harv|Gorenstein|Lyons|Solomon|1996|p=89}}.
 
=== Focal subgroup ===
If one defines, as in {{harv|Gorenstein|1980|p=246}}, the '''focal subgroup''' of ''P'' in ''G'' as the intersection ''P''∩[''G'',''G''] of the Sylow ''p''-subgroup ''P'' of the finite group ''G'' with the [[derived subgroup]] [''G'',''G''] of ''G'', then the focal subgroup is clearly important as it is a Sylow ''p''-subgroup of the derived subgroup. However, more importantly, one gets the following result:
:There exists a normal subgroup ''K'' of ''G'' with ''G''/''K'' an [[abelian group|abelian]] [[p-group|''p''-group]] isomorphic to ''P''/''P''∩[''G'',''G''] (here ''K'' denotes '''A'''<sup>''p''</sup>(''G'')), and
:if ''K'' is a normal subgroup of ''G'' with ''G''/''K'' an abelian p-group, then ''P''∩[''G'',''G''] ≤ ''K'', and ''G''/''K'' is a homomorphic image of ''P''/''P''∩[''G'',''G''], {{harv|Gorenstein|1980|loc=Theorem 7.3.1, p. 90}}.
 
One can define, as in {{harv|Isaacs|2008|p=165}} the '''focal subgroup''' of ''H'' with respect to ''G'' as:
:Foc<sub>''G''</sub>(''H'') = ⟨ ''x''<sup>−1</sup> ''y'' {{!}} ''x'',''y'' in ''H'' and ''x'' is ''G''-conjugate to ''y'' ⟩.
This focal subgroup measures the extent to which elements of ''H'' fuse in ''G'', while the previous definition measured certain abelian ''p''-group homomorphic images of the group ''G''.  The content of the focal subgroup theorem is that these two definitions of focal subgroup are compatible.
 
== Statement of the theorem ==
 
The focal subgroup of a finite group ''X'' with Sylow ''p''-subgroup ''P'' is given by:
:''P''∩[''G'',''G''] = ''P''∩'''A'''<sup>''p''</sup>(''G'') = ''P''∩ker(''v'') = Foc<sub>''G''</sub>(''P'') = ⟨ ''x''<sup>−1</sup> ''y'' {{!}} ''x'',''y'' in ''P'' and ''x'' is ''G''-conjugate to ''y'' ⟩
where ''v'' is the transfer homomorphism from ''G'' to ''P''/[''P'',''P''], {{harv |Isaacs|2008|loc=Theorem 5.21, p. 165}}.
 
== History and generalizations ==
This connection between transfer and fusion is credited to {{harv|Higman|1958}},<ref>The focal subgroup theorem and/or the focal subgroup is due to {{harv|Higman|1958}} according to {{harv|Gorenstein|Lyons|Solomon|1996|p=90}}, {{harv|Rose|1978|p=255}}, {{harv|Suzuki|1986|p=141}}; however, the focal subgroup theorem as stated there and here is quite a bit older and already appears in textbook form in {{harv|Hall|1959|p=215}}. There and in {{harv|Puig|2000}} the ideas are credited to {{harv|Grün|1935}}; compare to {{harv|Grün|1935|loc=Satz 5}} in the special case of ''p''-normal groups, and the general result in Satz 9 which is in some sense a refinement of the focal subgroup theorem.</ref> where, in different language, the focal subgroup theorem was proved along with various generalizations.  The requirement that ''G''/''K'' be abelian was dropped, so that Higman also studied '''O'''<sup>''p''</sup>(''G'') and the [[nilpotent residual]] γ<sub>∞</sub>(''G''), as so called '''hyperfocal subgroups'''.  Higman also did not restrict to a single prime ''p'', but rather allowed ''π''-groups for sets of primes ''π'' and used [[Philip Hall]]'s theorem of [[Hall subgroup]]s in order to prove similar results about the transfer into Hall ''π''-subgroups; taking ''π'' = {''p''} a Hall ''π''-subgroup is a Sylow ''p''-subgroup, and the results of Higman are as presented above.
 
Interest in the hyperfocal subgroups was renewed by work of {{harv|Puig|2000}} in understanding the [[modular representation theory]] of certain well behaved blocks. The hyperfocal subgroup of ''P'' in ''G'' can defined as ''P''∩γ<sub>∞</sub>(''G'') that is, as a Sylow ''p''-subgroup of the nilpotent residual of ''G''.  If ''P'' is a Sylow ''p''-subgroup of the finite group ''G'', then one gets the standard focal subgroup theorem:
:''P''∩γ<sub>∞</sub>(''G'') = ''P''∩'''O'''<sup>''p''</sup>(''G'') = ⟨ ''x''<sup>−1</sup> ''y'' : ''x'',''y'' in ''P'' and ''y'' = ''x''<sup>''g''</sup> for some ''g'' in ''G'' of order coprime to ''p'' ⟩
and the local characterization:
:''P''∩'''O'''<sup>''p''</sup>(''G'') = ⟨ ''x''<sup>−1</sup> ''y'' : ''x'',''y'' in ''Q'' ≤ ''P'' and ''y'' = ''x''<sup>''g''</sup> for some ''g'' in N<sub>''G''(''Q'')</sub> of order coprime to ''p'' ⟩.
This compares to the local characterization of the focal subgroup as:
:''P''∩'''A'''<sup>''p''</sup>(''G'') = ⟨ ''x''<sup>−1</sup> ''y'' : ''x'',''y'' in ''Q'' ≤ ''P'' and ''y'' = ''x''<sup>''g''</sup> for some ''g'' in N<sub>''G''</sub>(''Q'') ⟩.
 
Puig is interested in the generalization of this situation to '''fusion systems''', a [[category theory|categorical]] model of the fusion pattern of a Sylow ''p''-subgroup with respect to a finite group that also models the fusion pattern of a defect group of a ''p''-block in modular representation theory.  In fact fusion systems have found a number of surprising applications and inspirations in the area of [[algebraic topology]] known as [[equivariant cohomology|equivariant]] [[homotopy theory]]. Some of the major algebraic theorems in this area only have topological proofs at the moment.
 
== Other characterizations ==
 
Various mathematicians have presented methods to calculate the focal subgroup from smaller groups.  For instance, the influential work {{harv|Alperin|1967}} develops the idea of a local control of fusion, and as an example application shows that:
:''P''&nbsp;∩&nbsp;'''A'''<sup>''p''</sup>(''G'') is generated by the commutator subgroups [''Q'', N<sub>''G''</sub>(''Q'')] where ''Q'' varies over a family ''C'' of subgroups of&nbsp;''P''
The choice of the family ''C'' can be made in many ways (''C'' is what is called a "weak conjugation family" in {{harv|Alperin|1967}}), and several examples are given: one can take ''C'' to be all non-identity subgroups of ''P'', or the smaller choice of just the intersections ''Q''&nbsp;=&nbsp;''P''&nbsp;∩&nbsp;''P''<sup>''g''</sup> for ''g'' in ''G'' in which N<sub>''P''</sub>(''Q'') and N<sub>''P''<sup>''g''</sup></sub>(''Q'') are both Sylow ''p''-subgroups of N<sub>''G''</sub>(''Q'').  The latter choice is made in {{harv|Gorenstein|1980|loc=Theorem 7.4.1, p. 251}}.  The work of {{harv|Grün|1935}} studied aspects of the transfer and fusion as well, resulting in '''Grün's first theorem''':
:''P''&nbsp;∩&nbsp;'''A'''<sup>''p''</sup>(''G'') is generated by ''P''∩[''N'',''N''] and ''P''&nbsp;∩&nbsp;[''Q'',&nbsp;''Q''] where ''N''&nbsp;=&nbsp;N<sub>''G''</sub>(''P'') and ''Q'' ranges over the set of Sylow ''p''-subgroups ''Q'' = ''P''<sup>''g''</sub> of ''G'' {{harv|Gorenstein|1980|loc=Theorem 7.4.2, p. 252}}.
 
== Applications ==
 
{{Expand section|date=January 2010|suggestion=Control of transfer leads to various p-nilpotence results; include Tate and Thompsons hyperfocal O^p and "hypo" focal E^p results; include the character theoretic transfer and the work of Yoshida; contrast to control of fusion of elements and to control of strong fusion}}
 
The textbook presentations in {{harv|Rose|1978|pp=254–264}}, {{harv|Isaacs|2008|loc=Chapter 5}}, {{harv|Hall|1959|loc=Chapter 14}}, {{harv|Suzuki|1986|loc=&sect;5.2, pp. 138–165}}, all contain various applications of the focal subgroup theorem relating fusion, transfer, and a certain kind of [[split exact sequence|splitting]] called [[p-nilpotent group|''p''-nilpotence]].
 
During the course of the [[Alperin–Brauer–Gorenstein theorem]] classifying finite [[simple group]]s with [[quasi-dihedral group|quasi-dihedral]] Sylow 2-subgroups, it becomes necessary to distinguish four types of groups with quasi-dihedral Sylow 2-subgroups: the 2-nilpotent groups, the ''Q''-type groups whose focal subgroup is a [[generalized quaternion group]] of index 2, the ''D''-type groups whose focal subgroup a [[dihedral group]] of index 2, and the ''QD''-type groups whose focal subgroup is the entire quasi-dihedral group.  In terms of fusion, the 2-nilpotent groups have 2 classes of involutions, and 2 classes of cyclic subgroups of order 4; the ''Q''-type have 2 classes of involutions and one class of cyclic subgroup of order 4; the ''QD''-type have one class each of involutions and cyclic subgroups of order 4.  In other words, finite groups with quasi-dihedral Sylow 2-subgroups can be classified according to their focal subgroup, or equivalently, according to their fusion patterns.  The explicit lists of groups with each fusion pattern are contained in {{harv|Alperin|Brauer|Gorenstein|1970}}.
 
== Notes ==
<references/>
 
== References ==
* {{Citation | last1=Alperin | first1=J. L. | author1-link=J. L. Alperin | title=Sylow intersections and fusion | doi=10.1016/0021-8693(67)90005-1 | mr=0215913 | year=1967 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=6 | pages=222–241}}
* {{Citation | last1=Alperin | first1=J. L. | author1-link=J. L. Alperin | last2=Brauer | first2=R. | author2-link=Richard Brauer | last3=Gorenstein | first3=D. | author3-link=Daniel Gorenstein | title=Finite groups with quasi-dihedral and wreathed Sylow 2-subgroups. | doi=10.2307/1995627 | mr=0284499 | year=1970 | journal=[[Transactions of the American Mathematical Society]] | publisher=[[American Mathematical Society]] | issn=0002-9947 | volume=151 | pages=1–261}}
* {{Citation | last1=Gorenstein | first1=D. | author1-link=Daniel Gorenstein | title=Finite Groups | publisher=Chelsea | location=New York | isbn=978-0-8284-0301-6 | mr=81b:20002 | year=1980}}
* {{Citation | last1=Gorenstein | first1=D. | author1-link=Daniel Gorenstein | last2=Lyons | first2=Richard | last3=Solomon | first3=Ronald | title=The classification of the finite simple groups. Number 2. Part I. Chapter G | url=https://www.ams.org/online_bks/surv402 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-0390-5 | mr=1358135 | year=1996 | volume=40}}
* {{Citation | last1=Grün | first1=Otto | title=Beiträge zur Gruppentheorie. I. | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002173409 | language=German | zbl=0012.34102 | year=1935 | journal=Journal für Reine und Angewandte Mathematik | issn=0075-4102 | volume=174 | pages=1–14}}
* {{Citation | last1=Hall | first1=Marshall, Jr. | author1-link=Marshall Hall (mathematician) | title=The theory of groups | publisher=Macmillan | location=New York | mr=0103215 | year=1959}}
* {{Citation | last1=Higman | first1=Donald G. | author1-link=Donald G. Higman | title=Focal series in finite groups | url=http://www.cms.math.ca/cjm/v5/p477 | mr=0058597 | year=1953 | journal=[[Canadian Journal of Mathematics]] | issn=0008-414X | volume=5 | pages=477–497}}
* {{Citation | last1=Puig | first1=Lluis | title=The hyperfocal subalgebra of a block | doi=10.1007/s002220000072 | mr=1775217 | year=2000 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=141 | issue=2 | pages=365–397}}
* {{Citation | last1=Rose | first1=John S. | title=A Course in Group Theory | origyear=1978 | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-68194-8 | mr=0498810 | year=1994}}
* {{Citation | last1=Suzuki | first1=Michio | author1-link=Michio Suzuki | title=Group theory. II | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-0-387-10916-9 | mr=815926 | year=1986 | volume=248}}
 
{{DEFAULTSORT:Focal Subgroup Theorem}}
[[Category:P-groups]]
[[Category:Theorems in group theory]]

Latest revision as of 04:31, 10 December 2013

In abstract algebra, the focal subgroup theorem describes the fusion of elements in a Sylow subgroup of a finite group. The focal subgroup theorem was introduced in Template:Harv and is the "first major application of the transfer" according to Template:Harv. The focal subgroup theorem relates the ideas of transfer and fusion such as described in Template:Harv. Various applications of these ideas include local criteria for p-nilpotence and various non-simplicity criterion focussing on showing that a finite group has a normal subgroup of index p.

Background

The focal subgroup theorem relates several lines of investigation in finite group theory: normal subgroups of index a power of p, the transfer homomorphism, and fusion of elements.

Subgroups

The following three normal subgroups of index a power of p are naturally defined, and arise as the smallest normal subgroups such that the quotient is (a certain kind of) p-group. Formally, they are kernels of the reflection onto the reflective subcategory of p-groups (respectively, elementary abelian p-groups, abelian p-groups).

  • Ep(G) is the intersection of all index p normal subgroups; G/Ep(G) is an elementary abelian group, and is the largest elementary abelian p-group onto which G surjects.
  • Ap(G) (notation from Template:Harv) is the intersection of all normal subgroups K such that G/K is an abelian p-group (i.e., K is an index pk normal subgroup that contains the derived group [G,G]): G/Ap(G) is the largest abelian p-group (not necessarily elementary) onto which G surjects.
  • Op(G) is the intersection of all normal subgroups K of G such that G/K is a (possibly non-abelian) p-group (i.e., K is an index pk normal subgroup): G/Op(G) is the largest p-group (not necessarily abelian) onto which G surjects. Op(G) is also known as the <p-residual subgroup>...</p-residual subgroup>p-residual subgroup.

Firstly, as these are weaker conditions on the groups K, one obtains the containments Ep(G)⊇Ap(G)⊇Op(G). These are further related as:

Ap(G) = Op(G)[G,G].

Op(G) has the following alternative characterization as the subgroup generated by all Sylow q-subgroups of G as q≠p ranges over the prime divisors of the order of G distinct from p.

Op(G) is used to define the lower p-series<lower p-series>...</lower p-series> of G, similarly to the upper p-series described in p-core.

Transfer homomorphism

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. The transfer homomorphism is a homomorphism that can be defined from any group G to the abelian group H/[H,H] defined by a subgroup H ≤ G of finite index, that is [G:H] < ∞. The transfer map from a finite group G into its Sylow p-subgroup has a kernel that is easy to describe:

The kernel of the transfer homomorphism from a finite group G into its Sylow p-subgroup P has Ap(G) as its kernel, Template:Harv.

In other words, the "obvious" homomorphism onto an abelian p-group is in fact the most general such homomorphism.

Fusion

The fusion pattern of a subgroup H in G is the equivalence relation on the elements of H where two elements h, k of H are fused if they are G-conjugate, that is, if there is some g in G such that h = kg. The normal structure of G has an effect on the fusion pattern of its Sylow p-subgroups, and conversely the fusion pattern of its Sylow p-subgroups has an effect on the normal structure of G, Template:Harv.

Focal subgroup

If one defines, as in Template:Harv, the focal subgroup of P in G as the intersection P∩[G,G] of the Sylow p-subgroup P of the finite group G with the derived subgroup [G,G] of G, then the focal subgroup is clearly important as it is a Sylow p-subgroup of the derived subgroup. However, more importantly, one gets the following result:

There exists a normal subgroup K of G with G/K an abelian p-group isomorphic to P/P∩[G,G] (here K denotes Ap(G)), and
if K is a normal subgroup of G with G/K an abelian p-group, then P∩[G,G] ≤ K, and G/K is a homomorphic image of P/P∩[G,G], Template:Harv.

One can define, as in Template:Harv the focal subgroup of H with respect to G as:

FocG(H) = ⟨ x−1 y | x,y in H and x is G-conjugate to y ⟩.

This focal subgroup measures the extent to which elements of H fuse in G, while the previous definition measured certain abelian p-group homomorphic images of the group G. The content of the focal subgroup theorem is that these two definitions of focal subgroup are compatible.

Statement of the theorem

The focal subgroup of a finite group X with Sylow p-subgroup P is given by:

P∩[G,G] = P∩Ap(G) = P∩ker(v) = FocG(P) = ⟨ x−1 y | x,y in P and x is G-conjugate to y ⟩

where v is the transfer homomorphism from G to P/[P,P], Template:Harv.

History and generalizations

This connection between transfer and fusion is credited to Template:Harv,[1] where, in different language, the focal subgroup theorem was proved along with various generalizations. The requirement that G/K be abelian was dropped, so that Higman also studied Op(G) and the nilpotent residual γ∞(G), as so called hyperfocal subgroups. Higman also did not restrict to a single prime p, but rather allowed π-groups for sets of primes π and used Philip Hall's theorem of Hall subgroups in order to prove similar results about the transfer into Hall π-subgroups; taking π = {p} a Hall π-subgroup is a Sylow p-subgroup, and the results of Higman are as presented above.

Interest in the hyperfocal subgroups was renewed by work of Template:Harv in understanding the modular representation theory of certain well behaved blocks. The hyperfocal subgroup of P in G can defined as P∩γ∞(G) that is, as a Sylow p-subgroup of the nilpotent residual of G. If P is a Sylow p-subgroup of the finite group G, then one gets the standard focal subgroup theorem:

P∩γ∞(G) = P∩Op(G) = ⟨ x−1 y : x,y in P and y = xg for some g in G of order coprime to p ⟩

and the local characterization:

P∩Op(G) = ⟨ x−1 y : x,y in Q ≤ P and y = xg for some g in NG(Q) of order coprime to p ⟩.

This compares to the local characterization of the focal subgroup as:

P∩Ap(G) = ⟨ x−1 y : x,y in Q ≤ P and y = xg for some g in NG(Q) ⟩.

Puig is interested in the generalization of this situation to fusion systems, a categorical model of the fusion pattern of a Sylow p-subgroup with respect to a finite group that also models the fusion pattern of a defect group of a p-block in modular representation theory. In fact fusion systems have found a number of surprising applications and inspirations in the area of algebraic topology known as equivariant homotopy theory. Some of the major algebraic theorems in this area only have topological proofs at the moment.

Other characterizations

Various mathematicians have presented methods to calculate the focal subgroup from smaller groups. For instance, the influential work Template:Harv develops the idea of a local control of fusion, and as an example application shows that:

P ∩ Ap(G) is generated by the commutator subgroups [Q, NG(Q)] where Q varies over a family C of subgroups of P

The choice of the family C can be made in many ways (C is what is called a "weak conjugation family" in Template:Harv), and several examples are given: one can take C to be all non-identity subgroups of P, or the smaller choice of just the intersections Q = P ∩ Pg for g in G in which NP(Q) and NPg(Q) are both Sylow p-subgroups of NG(Q). The latter choice is made in Template:Harv. The work of Template:Harv studied aspects of the transfer and fusion as well, resulting in Grün's first theorem:

P ∩ Ap(G) is generated by P∩[N,N] and P ∩ [Q, Q] where N = NG(P) and Q ranges over the set of Sylow p-subgroups Q = Pg of G Template:Harv.

Applications

Template:Expand section

The textbook presentations in Template:Harv, Template:Harv, Template:Harv, Template:Harv, all contain various applications of the focal subgroup theorem relating fusion, transfer, and a certain kind of splitting called p-nilpotence.

During the course of the Alperin–Brauer–Gorenstein theorem classifying finite simple groups with quasi-dihedral Sylow 2-subgroups, it becomes necessary to distinguish four types of groups with quasi-dihedral Sylow 2-subgroups: the 2-nilpotent groups, the Q-type groups whose focal subgroup is a generalized quaternion group of index 2, the D-type groups whose focal subgroup a dihedral group of index 2, and the QD-type groups whose focal subgroup is the entire quasi-dihedral group. In terms of fusion, the 2-nilpotent groups have 2 classes of involutions, and 2 classes of cyclic subgroups of order 4; the Q-type have 2 classes of involutions and one class of cyclic subgroup of order 4; the QD-type have one class each of involutions and cyclic subgroups of order 4. In other words, finite groups with quasi-dihedral Sylow 2-subgroups can be classified according to their focal subgroup, or equivalently, according to their fusion patterns. The explicit lists of groups with each fusion pattern are contained in Template:Harv.

Notes

  1. ↑ The focal subgroup theorem and/or the focal subgroup is due to Template:Harv according to Template:Harv, Template:Harv, Template:Harv; however, the focal subgroup theorem as stated there and here is quite a bit older and already appears in textbook form in Template:Harv. There and in Template:Harv the ideas are credited to Template:Harv; compare to Template:Harv in the special case of p-normal groups, and the general result in Satz 9 which is in some sense a refinement of the focal subgroup theorem.

References

  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
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