Socle (mathematics)

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 u4 (which gives a normal subgroup with 3 elements) and the other by u6 (which gives a normal subgroup with 2 elements). 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,

:\mathrm{soc}(M) = \sum_{N \text{ is a simple submodule of }M} N. Equivalently, :\mathrm{soc}(M) = \bigcap_{E \text{ is an essential submodule of }M} E.

The socle of a ring R can refer to one of two sets in the ring. Considering R as a right R-module, \mathrm{soc}(R_R) is defined, and considering R as a left R-module, \mathrm{soc}(_RR) is defined. Both of these socles are ring ideals, and it is known they are not necessarily equal.

• If M is an Artinian module, \mathrm{soc}(M) is itself an essential submodule of M. In fact, if M is a semiartinian module, then \mathrm{soc}(M) is itself an essential submodule of M. Additionally, if M is a non-zero module over a left semi-Artinian ring, then \mathrm{soc}(M) is itself an essential submodule of M. This is because any non-zero module over a left semi-Artinian ring is a semiartinian module.
• A module is semisimple if and only if \mathrm{soc}(M)=M. Rings for which \mathrm{soc}(M)=M for all module M are precisely semisimple rings.
• \mathrm{soc}(\mathrm{soc}(M))=\mathrm{soc}(M).
• M is a finitely cogenerated module if and only if \mathrm{soc}(M) is finitely generated and \mathrm{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 semisimple submodule.
• From the definition of \mathrm{rad}(R), it is easy to see that \mathrm{rad}(R) annihilates \mathrm{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.

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.)

References

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