Socle (mathematics): Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
en>Rschwieb
(boldfacing terms which probably redirect here)
 
en>Yobot
m (→‎Socle of a group: Reference before punctuation using AWB (9585))
Line 2: Line 2:


==Socle of a group==
==Socle of a group==
In the context of [[group theory]], the '''socle of a [[group (mathematics)|group]]''' ''G'', denoted soc(''G''), is the [[subgroup]] generated by the [[minimal normal subgroup]]s of ''G''. It can happen that a group has no minimal non-trivial normal subgroup (that is, every non-trivial normal subgroup properly contains another such subgroup) and in that case the socle is defined to be the subgroup generated by the identity. The socle is a direct product of minimal normal subgroups{{sfn|Robinson|1996|loc=p.87}}
In the context of [[group theory]], the '''socle of a [[group (mathematics)|group]]''' ''G'', denoted soc(''G''), is the [[subgroup]] generated by the [[minimal normal subgroup]]s of ''G''. It can happen that a group has no minimal non-trivial normal subgroup (that is, every non-trivial normal subgroup properly contains another such subgroup) and in that case the socle is defined to be the subgroup generated by the identity. The socle is a direct product of minimal normal subgroups.{{sfn|Robinson|1996|loc=p.87}}


As an example, consider the [[cyclic group]] '''Z'''<sub>12</sub> with [[generating set of a group|generator]] ''u'', which has two minimal normal subgroups, one generated by ''u''<sup> 4</sup> (which gives a normal subgroup with 3 elements) and the other by ''u''<sup> 6</sup> (which gives a normal subgroup with 2 elements). Thus the socle of '''Z'''<sub>12</sub> is the group generated by ''u''<sup> 4</sup> and ''u''<sup> 6</sup>, which is just the group generated by ''u''<sup> 2</sup>.  
As an example, consider the [[cyclic group]] '''Z'''<sub>12</sub> with [[generating set of a group|generator]] ''u'', which has two minimal normal subgroups, one generated by ''u''<sup> 4</sup> (which gives a normal subgroup with 3 elements) and the other by ''u''<sup> 6</sup> (which gives a normal subgroup with 2 elements). Thus the socle of '''Z'''<sub>12</sub> is the group generated by ''u''<sup> 4</sup> and ''u''<sup> 6</sup>, which is just the group generated by ''u''<sup> 2</sup>.


The socle is a [[characteristic subgroup]], and hence a normal subgroup. It is not necessarily [[Transitively normal subgroup|transitively normal]], however.  
The socle is a [[characteristic subgroup]], and hence a normal subgroup. It is not necessarily [[Transitively normal subgroup|transitively normal]], however.


If a group G is a finite [[solvable group]], then the socle is can be expressed as a product of [[elementary abelian]] [[p-group|p-groups]]. Thus, in this case, it just a product of copies of '''Z/pZ''' for various '''p''' where the same '''p''' may occur multiple times in the product.
If a group G is a finite [[solvable group]], then the socle is can be expressed as a product of [[elementary abelian]] [[p-group]]s. Thus, in this case, it just a product of copies of '''Z/pZ''' for various '''p''' where the same '''p''' may occur multiple times in the product.


==Socle of a module==
==Socle of a module==
In the context of [[module theory]] and [[ring theory]] the '''socle of a [[module (mathematics)|module]]''' ''M'' over a [[ring (mathematics)|ring]] ''R'' is defined to be the sum of the minimal nonzero submodules of ''M''. It can be considered as a [[duality (mathematics)|dual notion]] to that of the [[radical of a module]]. In set notation,
In the context of [[module theory]] and [[ring theory]] the '''socle of a [[module (mathematics)|module]]''' ''M'' over a [[ring (mathematics)|ring]] ''R'' is defined to be the sum of the minimal nonzero submodules of ''M''. It can be considered as a [[duality (mathematics)|dual notion]] to that of the [[radical of a module]]. In set notation,


:<math>\mathrm {soc}(M) = \sum \{ N \mid N \mbox{ is a simple submodule of M} \} \,</math>
:<math>\mathrm {soc}(M) = \sum \{ N \mid N \text{ is a simple submodule of }M \}. \,</math>
Equivalently,
Equivalently,
:<math>\mathrm {soc}(M) = \bigcap\{ E \mid E \mbox{ is an essential submodule of M} \} \,</math>
:<math>\mathrm {soc}(M) = \bigcap\{ E \mid E \text{ is an essential submodule of }M \}. \,</math>


The '''socle of a ring''' ''R'' can refer to one of two sets in the ring. Considering ''R'' as a right ''R'' module, soc(''R''<sub>''R''</sub>) is defined, and considering ''R'' as a left ''R'' module, soc(<sub>''R''</sub>''R'') is defined. Both of these socles are [[ring ideal]]s, and it is known they are not necessarily equal.  
The '''socle of a ring''' ''R'' can refer to one of two sets in the ring. Considering ''R'' as a right ''R'' module, soc(''R''<sub>''R''</sub>) is defined, and considering ''R'' as a left ''R'' module, soc(<sub>''R''</sub>''R'') is defined. Both of these socles are [[ring ideal]]s, and it is known they are not necessarily equal.


* If ''M'' is an [[Artinian module]], soc(''M'') is itself an essential submodule of ''M''.
* If ''M'' is an [[Artinian module]], soc(''M'') is itself an [[essential extension|essential submodule]] of ''M''.
* A module is [[semisimple module|semisimple]] if and only if soc(''M'') = ''M''.  Rings for which soc(''M'')=''M'' for all ''M'' are precisely [[semisimple ring]]s.
* A module is [[semisimple module|semisimple]] if and only if soc(''M'')&nbsp;=&nbsp;''M''.  Rings for which soc(''M'')&nbsp;=&nbsp;''M'' for all ''M'' are precisely [[semisimple ring]]s.
*M is a [[finitely cogenerated module]] if and only if soc(''M'') is finitely generated and rad(M) is an essential submodule of M.  
*M is a [[finitely cogenerated module]] if and only if soc(''M'') is finitely generated and soc(M) is an [[essential extension|essential submodule]] of M.  
*Since the sum of semisimple modules is semisimple, the socle of a module could also be defined as the unique maximal semi-simple submodule.  
*Since the sum of semisimple modules is semisimple, the socle of a module could also be defined as the unique maximal semi-simple submodule.  
* From the definition of rad(''R''), it is easy to see that rad(''R'') [[annihilator (ring theory)|annihilates]] soc(''R''). If ''R'' is a finite dimensional unital [[algebra]] and ''M'' a finitely generated ''R''-module then the socle consists precisely of the elements annihilated by the [[Jacobson radical]] of ''R''.<ref>[[J. L. Alperin]]; Rowen B. Bell, ''Groups and Representations'', 1995,  ISBN 0-387-94526-1, p. 136</ref>
* From the definition of rad(''R''), it is easy to see that rad(''R'') [[annihilator (ring theory)|annihilates]] soc(''R''). If ''R'' is a finite dimensional unital [[algebra]] and ''M'' a finitely generated ''R''-module then the socle consists precisely of the elements annihilated by the [[Jacobson radical]] of ''R''.<ref>[[J. L. Alperin]]; Rowen B. Bell, ''Groups and Representations'', 1995,  ISBN 0-387-94526-1, p. 136</ref>


==Socle of a Lie algebra==
==Socle of a Lie algebra==
In the context of [[Lie algebra]]s, a '''socle of a [[symmetric Lie algebra]]''' is the [[eigenspace]] of its structural [[automorphism]] which corresponds to the eigenvalue -1. (A symmetric Lie algebra decomposes into the [[direct sum of modules|direct sum]] of its socle and [[cosocle]].)<ref>[[Mikhail Postnikov]], ''Geometry VI: Riemannian Geometry'', 2001,  ISBN 3540411089,[http://books.google.com/books?id=P60o2UKOaPcC&pg=PA98&dq=cosocle#v=onepage&q=socle&f=false p. 98]</ref>
In the context of [[Lie algebra]]s, a '''socle of a [[symmetric Lie algebra]]''' is the [[eigenspace]] of its structural [[automorphism]] which corresponds to the eigenvalue&nbsp;&minus;1. (A symmetric Lie algebra decomposes into the [[direct sum of modules|direct sum]] of its socle and [[cosocle]].)<ref>[[Mikhail Postnikov]], ''Geometry VI: Riemannian Geometry'', 2001,  ISBN 3540411089,[http://books.google.com/books?id=P60o2UKOaPcC&pg=PA98&dq=cosocle#v=onepage&q=socle&f=false p. 98]</ref>


==See also==
==See also==
Line 40: Line 39:
* {{cite book | last1=Anderson | first1=Frank Wylie | last2=Fuller|first2=Kent R. | title=Rings and Categories of Modules | publisher=[[Springer-Verlag]] | isbn=978-0-387-97845-1 | year=1992}}
* {{cite book | last1=Anderson | first1=Frank Wylie | last2=Fuller|first2=Kent R. | title=Rings and Categories of Modules | publisher=[[Springer-Verlag]] | isbn=978-0-387-97845-1 | year=1992}}


*{{citation |last1=Robinson |first1=Derek J. S. |title=A course in the theory of groups |series=Graduate Texts in Mathematics |volume=80 |edition=2  |publisher=[[Springer-Verlag]]  |place=New York |year=1996 |pages=xviii+499 |isbn=0-387-94461-3 |MR=1357169}}
*{{citation |last1=Robinson |first1=Derek J. S. |title=A course in the theory of groups |series=Graduate Texts in Mathematics |volume=80 |edition=2  |publisher=[[Springer-Verlag]]  |place=New York |year=1996 |pages=xviii+499 |isbn=0-387-94461-3 |mr=1357169}}


[[Category:Module theory]]
[[Category:Module theory]]
[[Category:Group theory]]
[[Category:Group theory]]
[[Category:Functional subgroups]]
[[Category:Functional subgroups]]
{{Abstract-algebra-stub}}
[[uk:Цоколь (алгебра)]]

Revision as of 15:44, 7 November 2013

In mathematics, the term socle has several related meanings.

Socle of a group

In the context of group theory, the socle of a group G, denoted soc(G), is the subgroup generated by the minimal normal subgroups of G. It can happen that a group has no minimal non-trivial normal subgroup (that is, every non-trivial normal subgroup properly contains another such subgroup) and in that case the socle is defined to be the subgroup generated by the identity. The socle is a direct product of minimal normal subgroups.Template:Sfn

As an example, consider the cyclic group Z12 with generator u, which has two minimal normal subgroups, one generated by u 4 (which gives a normal subgroup with 3 elements) and the other by u 6 (which gives a normal subgroup with 2 elements). Thus the socle of Z12 is the group generated by u 4 and u 6, which is just the group generated by u 2.

The socle is a characteristic subgroup, and hence a normal subgroup. It is not necessarily transitively normal, however.

If a group G is a finite solvable group, then the socle is can be expressed as a product of elementary abelian p-groups. Thus, in this case, it just a product of copies of Z/pZ for various p where the same p may occur multiple times in the product.

Socle of a module

In the context of module theory and ring theory the socle of a module M over a ring R is defined to be the sum of the minimal nonzero submodules of M. It can be considered as a dual notion to that of the radical of a module. In set notation,

Equivalently,

The socle of a ring R can refer to one of two sets in the ring. Considering R as a right R module, soc(RR) is defined, and considering R as a left R module, soc(RR) is defined. Both of these socles are ring ideals, and it is known they are not necessarily equal.

Socle of a Lie algebra

In the context of Lie algebras, a socle of a symmetric Lie algebra is the eigenspace of its structural automorphism which corresponds to the eigenvalue −1. (A symmetric Lie algebra decomposes into the direct sum of its socle and cosocle.)[2]

See also

References

  1. J. L. Alperin; Rowen B. Bell, Groups and Representations, 1995, ISBN 0-387-94526-1, p. 136
  2. Mikhail Postnikov, Geometry VI: Riemannian Geometry, 2001, ISBN 3540411089,p. 98
  • {{#invoke:citation/CS1|citation

|CitationClass=book }}

  • {{#invoke:citation/CS1|citation

|CitationClass=book }}

  • {{#invoke:citation/CS1|citation

|CitationClass=citation }}