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, is the subgroup generated by the minimal normal subgroups of G. It can happen that a group has no minimal non-trivial normal 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.
As an example, consider the cyclic group Z12 with generator u, which has two minimal normal subgroups, one generated by u4 and the other by u6. Thus the socle of Z12 is the group generated by u4 and u6, which is just the group generated by u2.
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 is defined, and considering R as a left R module, soc is defined. Both of these socles are ring ideals, and it is known they are not necessarily equal.
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.