In mathematics, the term socle has several related meanings.
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 Z<sub>12</sub> with generator u, which has two minimal normal subgroups, one generated by u<sup>4</sup> (which gives a normal subgroup with 3 elements) and the other by u<sup>6</sup> (which gives a normal subgroup with 2 elements). Thus the socle of Z<sub>12</sub> is the group generated by u<sup>4</sup> and u<sup>6</sup>, which is just the group generated by u<sup>2</sup>.
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.
In the context of module theory and ring theory the socle of a module over a ring is defined to be the sum of the minimal nonzero submodules of . 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 can refer to one of two sets in the ring. Considering as a right -module, is defined, and considering as a left -module, is defined. Both of these socles are ring ideals, and it is known they are not necessarily equal.
In fact, if is a semiartinian module, then is itself an essential submodule of . Additionally, if is a non-zero module over a left semi-Artinian ring, then is itself an essential submodule of . This is because any non-zero module over a left semi-Artinian ring is a semiartinian module.
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 âÂÂ1. (A symmetric Lie algebra decomposes into the direct sum of its socle and cosocle.)