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| In [[mathematics]], or more specifically [[group theory]], the '''omega''' and '''agemo''' subgroups described the so-called "power structure" of a [[finite group|finite]] [[p-group|''p''-group]]. They were introduced in {{harv|Hall|1933}} where they were used to describe a class of finite ''p''-groups whose structure was sufficiently similar to that of finite [[abelian group|abelian]] ''p''-groups, the so-called, [[regular p-group]]s. The relationship between power and commutator structure forms a central theme in the modern study of ''p''-groups, as exemplified in the work on uniformly [[powerful p-group]]s.
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| == Definition ==
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| The omega subgroups are the series of subgroups of a finite p-group, ''G'', indexed by the natural numbers:
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| :<math>\Omega_i(G) = \langle \{g : g^{p^i} = 1 \} \rangle. </math> <!-- Ω -->
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| The agemo subgroups are the series of subgroups:
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| :<math> \mho^i(G) = \langle \{ g^{p^i} : g \in G \} \rangle. </math> <!-- ℧ -->
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| When ''i'' = 1 and ''p'' is odd, then ''i'' is normally omitted from the definition. When ''p'' is even, an omitted ''i'' may mean either ''i'' = 1 or ''i'' = 2 depending on local convention. In this article, we use the convention that an omitted ''i'' always indicates ''i'' = 1.
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| == Examples ==
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| The [[dihedral group of order 8]], ''G'', satisfies: ℧(''G'') = Z(''G'') = [ ''G'', ''G'' ] = Φ(''G'') = Soc(''G'') is the unique normal subgroup of order 2, typically realized as the subgroup containing the identity and a 180° rotation. However Ω(''G'') = ''G'' is the entire group, since ''G'' is generated by reflections. This shows that Ω(''G'') need not be the set of elements of order ''p''.
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| The [[quaternion group of order 8]], ''H'', satisfies Ω(''H'') = ℧(''H'') = Z(''H'') = [ ''H'', ''H'' ] = Φ(''H'') = Soc(''H'') is the unique subgroup of order 2, normally realized as the subgroup containing only 1 and −1.
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| The [[Sylow subgroup|Sylow ''p''-subgroup]], ''P'', of the [[symmetric group]] on ''p''<sup>2</sup> points is the [[wreath product]] of two [[cyclic group]]s of prime order. When ''p'' = 2, this is just the dihedral group of order 8. It too satisfies Ω(''P'') = ''P''. Again ℧(''P'') = Z(''P'') = Soc(''P'') is cyclic of order ''p'', but [ ''P'', ''P'' ] = Φ(''G'') is elementary abelian of order ''p''<sup>''p''−1</sup>.
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| The [[semidirect product]] of a cyclic group of order 4 acting non-trivially on a cyclic group of order 4,
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| :<math> K = \langle a,b : a^4 = b^4 = 1, ba=ab^3 \rangle,</math>
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| has ℧(''K'') elementary abelian of order 4, but the set of squares is simply { 1, ''aa'', ''bb'' }. Here the element ''aabb'' of ℧(''K'') is not a square, showing that ℧ is not simply the set of squares.
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| ==Properties==
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| In this section, let ''G'' be a finite ''p''-group of [[order (group theory)|order]] |''G''| = ''p''<sup>''n''</sup> and [[exponent (group theory)|exponent]] exp(''G'') = ''p''<sup>''k''</sup> have a number of useful properties.
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| ;General properties:
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| *Both Ω<sub>''i''</sub>(''G'') and ℧<sup>i</sup>(''G'') are [[characteristic subgroup]]s of ''G'' for all natural numbers, ''i''.
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| * The omega and agemo subgroups form two [[normal series]]:
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| ::''G'' = ℧<sup>0</sup>(''G'') ≥ ℧<sup>1</sup>(''G'') ≥ ℧<sup>2</sup>(''G'') ≥ ... ≥ ℧<sup>''k''−2</sup>(''G'') ≥ ℧<sup>''k''−1</sup>(''G'') > ℧<sup>''k''</sup>(''G'') = 1
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| ::''G'' = Ω<sub>''k''</sub>(''G'') ≥ Ω<sub>''k''−1</sub>(''G'') ≥ Ω<sub>''k''−2</sub>(''G'') ≥ ... ≥ Ω<sub>2</sub>(''G'') ≥ Ω<sub>1</sup>(''G'') > Ω<sub>0</sub>(''G'') = 1
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| :and the series are loosely intertwined: For all ''i'' between 1 and ''k'':
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| :: ℧<sup>''i''</sup>(''G'') ≤ Ω<sub>''k''−''i''</sub>(''G''), but
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| :: ℧<sup>''i''−1</sup>(''G'') is not contained in Ω<sub>''k''−''i''</sub>(''G''). <!-- {{Harv|McKay|2000|p=32}} -->
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| ;Behavior under quotients and subgroups:
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| If ''H'' ≤ ''G'' is a [[subgroup]] of ''G'' and ''N'' ⊲ ''G'' is a [[normal subgroup]] of ''G'', then:
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| * ℧<sup>''i''</sup>(''H'') ≤ ''H'' ∩ ℧<sup>''i''</sup>(''G'')
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| * Ω<sub>''i''</sub>(''H'') = ''H'' ∩ Ω<sub>''i''</sub>(''G'')
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| * ℧<sup>''i''</sup>(''N'') ⊲ ''G''
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| * Ω<sub>''i''</sub>(''N'') ⊲ ''G''
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| * ℧<sup>''i''</sup>(''G''/''N'') = ℧<sup>''i''</sup>(''G'')''N''/''N''
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| * Ω<sub>''i''</sub>(''G''/''N'') ≥ Ω<sub>''i''</sub>(''G'')''N''/''N''
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| ;Relation to other important subgroups:
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| *Soc(''G'') = Ω(Z(''G'')), the subgroup consisting of central elements of order ''p'' is the [[socle (mathematics)|socle]], Soc(''G''), of ''G''
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| *''Φ''(''G'') = ℧(''G'')[''G'',''G''], the subgroup generated by all ''p''th powers and [[commutators]] is the [[Frattini subgroup]], Φ(''G''), of ''G''.
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| ;Relations in special classes of groups:
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| * In an abelian ''p''-group, or more generally in a regular ''p''-group:
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| :: |℧<sup>''i''</sup>(''G'')|⋅|Ω<sub>''i''</sub>(''G'')| = |''G''|
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| :: [℧<sup>''i''</sup>(''G''):℧<sup>''i''+1</sup>(''G'')] = [Ω<sub>''i''</sub>(''G''):Ω<sub>''i''+1</sub>(''G'')],
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| :where |''H''| is the [[order (group theory)|order]] of ''H'' and [''H'':''K''] = |''H''|/|''K''| denotes the [[index (group theory)|index]] of the subgroups ''K'' ≤ ''H''. <!-- {{Harv|McKay|2000|p=33}} -->
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| == Applications ==
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| The first application of the omega and agemo subgroups was to draw out the analogy of [[regular p-group|regular]] ''p''-groups with [[abelian group|abelian]] ''p''-groups in {{harv|Hall|1933}}.
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| Groups in which Ω(''G'') ≤ Z(''G'') were studied by [[John G. Thompson]] and have seen several more recent applications.
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| The dual notion, groups with [''G'',''G''] ≤ ℧(''G'') are called [[powerful p-groups]] and were introduced by [[Avinoam Mann]]. These groups were critical for the proof of the '''coclass conjectures''' which introduced an important way to understand the structure and classification of finite ''p''-groups.
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| == References ==
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| * {{citation | id={{MathSciNet | id=1152800}} | last1=Dixon | first1=J. D. | last2=du Sautoy | first2=M. P. F. | author2-link=Marcus du Sautoy | last3=Mann | first3=A. | last4=Segal | first4=D. | title=Analytic pro-p-groups | publisher=[[Cambridge University Press]] | year=1991 | isbn=0-521-39580-1}}
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| * {{Citation | last1=Hall | first1=Philip | author1-link=Philip Hall | title=A contribution to the theory of groups of prime-power order | year=1933 | journal=Proceedings of the London Mathematical Society | volume=36 | pages=29–95 | doi=10.1112/plms/s2-36.1.29}}
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| *{{Citation | last1=Leedham-Green | first1=C. R. | author1-link=Charles Leedham-Green | last2=McKay | first2=Susan | title=The structure of groups of prime power order | publisher=[[Oxford University Press]] | series=London Mathematical Society Monographs. New Series | isbn=978-0-19-853548-5 | id={{MathSciNet | id = 1918951}} | year=2002 | volume=27}}
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| * {{Citation | last1=McKay | first1=Susan | title=Finite p-groups | publisher=University of London | series=Queen Mary Maths Notes | isbn=978-0-902480-17-9 | id={{MathSciNet | id = 1802994}} | year=2000 | volume=18}}
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| {{DEFAULTSORT:Omega And Agemo Subgroup}}
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| [[Category:Finite groups]]
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