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.
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/"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.
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.)