# Socle (mathematics)

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.[1]

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**/*p***Z** 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. - soc(soc(
*M*)) = soc(*M*). *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*.[2]

## 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 that corresponds to the eigenvalue −1. (A symmetric Lie algebra decomposes into the direct sum of its socle and cosocle.)[3]

## See also

## References

- Robinson 1996, p.87.
- 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

- Alperin, J.L.; Bell, Rowen B. (1995).
*Groups and Representations*. Springer-Verlag. p. 136. ISBN 0-387-94526-1. - Anderson, Frank Wylie; Fuller, Kent R. (1992).
*Rings and Categories of Modules*. Springer-Verlag. ISBN 978-0-387-97845-1. - Robinson, Derek J. S. (1996),
*A course in the theory of groups*, Graduate Texts in Mathematics,**80**(2 ed.), New York: Springer-Verlag, pp. xviii+499, doi:10.1007/978-1-4419-8594-1, ISBN 0-387-94461-3, MR 1357169