# Socle (mathematics): Difference between revisions

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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 | If a group G is a finite [[solvable group]], then the socle can be expressed as a product of [[elementary abelian]] [[p-group]]s. Thus, in this case, it is 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== | ||

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*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. | *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== |

## Latest revision as of 21:14, 22 May 2014

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 can be expressed as a product of elementary abelian p-groups. Thus, in this case, it is 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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