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.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 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,

${\displaystyle \mathrm {soc} (M)=\sum \{N\mid N{\text{ is a simple submodule of }}M\}.\,}$

Equivalently,

${\displaystyle \mathrm {soc} (M)=\bigcap \{E\mid E{\text{ is an essential submodule of }}M\}.\,}$

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]

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
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