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,

:$\mathrm(M) = \sum_ N.$
Equivalently,
:$\mathrm(M) = \bigcap_ E.$

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.

* If ''M'' is an Artinian module, soc(''M'') is itself an essential submodule of ''M''.
In fact, if ''M'' is a semiartinian module, then soc(''M'') is itself an essential submodule of ''M''. Additionally, if ''M'' is a non-zero module over a left semi-Artinian ring, then 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 soc(''M'') = ''M''. Rings for which soc(''M'') = ''M'' for all ''M'' are precisely semisimple rings.
* soc(soc(''M'')) = soc(''M'').
*''M'' is a finitely cogenerated module if and only if soc(''M'') is finitely generated and 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 rad(''R''), it is easy to see that rad(''R'') annihilates 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.) Mikhail Postnikov, ''Geometry VI: Riemannian Geometry'', 2001,
p. 98
/ref>

See also

* Injective hull
*Radical of a module
* Cosocle

References

*
*
*

{{Set index article, mathematics

Module theory
Group theory
Functional subgroups