# Socle (mathematics): Difference between revisions

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==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 | 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 \ | :<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 \ | :<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'') = ''M''. Rings for which soc(''M'') = ''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 | *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 | 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> | ||

==See also== | ==See also== | ||

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* {{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 | | *{{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]] | ||

## 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 **Z**_{12} 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 **Z**_{12} 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(*R*_{R}) is defined, and considering *R* as a left *R* module, soc(_{R}*R*) is defined. Both of these socles are ring ideals, and it is known they are not necessarily equal.

- If
*M*is an Artinian module, soc(*M*) is itself an essential submodule of*M*. - A module is semisimple if and only if soc(
*M*) =*M*. Rings for which soc(*M*) =*M*for all*M*are precisely semisimple rings. - M is a finitely cogenerated module if and only if soc(
*M*) is finitely generated and soc(M) is an 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.
- From the definition of rad(
*R*), it is easy to see that rad(*R*) 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*.^{[1]}

## 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

- ↑ J. L. Alperin; Rowen B. Bell,
*Groups and Representations*, 1995, ISBN 0-387-94526-1, p. 136 - ↑ Mikhail Postnikov,
*Geometry VI: Riemannian Geometry*, 2001, ISBN 3540411089,p. 98

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