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 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 {\displaystyle M} over a ring R {\displaystyle R} is defined to be the sum of the minimal nonzero submodules of M {\displaystyle M} . It can be considered as a dual notion to that of the radical of a module. In set notation,
s o c ( M ) = ∑ N is a simple submodule of M N . {\displaystyle \mathrm {soc} (M)=\sum _{N{\text{ is a simple submodule of }}M}N.}
Equivalently,
s o c ( M ) = ⋂ E is an essential submodule of M E . {\displaystyle \mathrm {soc} (M)=\bigcap _{E{\text{ is an essential submodule of }}M}E.}
The socle of a ring R {\displaystyle R} can refer to one of two sets in the ring. Considering R {\displaystyle R} as a right R {\displaystyle R} -module, s o c ( R R ) {\displaystyle \mathrm {soc} (R_{R})} is defined, and considering R {\displaystyle R} as a left R {\displaystyle R} -module, s o c ( R R ) {\displaystyle \mathrm {soc} (_{R}R)} is defined. Both of these socles are ring ideals, and it is known they are not necessarily equal.
If M {\displaystyle M} is an Artinian module, s o c ( M ) {\displaystyle \mathrm {soc} (M)} is itself an essential submodule of M {\displaystyle M} . In fact, if M {\displaystyle M} is a semiartinian module, then s o c ( M ) {\displaystyle \mathrm {soc} (M)} is itself an essential submodule of M {\displaystyle M} . Additionally, if M {\displaystyle M} is a non-zero module over a left semi-Artinian ring, then s o c ( M ) {\displaystyle \mathrm {soc} (M)} is itself an essential submodule of M {\displaystyle 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 s o c ( M ) = M {\displaystyle \mathrm {soc} (M)=M} . Rings for which s o c ( M ) = M {\displaystyle \mathrm {soc} (M)=M} for all module M {\displaystyle M} are precisely semisimple rings.
s o c ( s o c ( M ) ) = s o c ( M ) {\displaystyle \mathrm {soc} (\mathrm {soc} (M))=\mathrm {soc} (M)} .
M {\displaystyle M} is a finitely cogenerated module if and only if s o c ( M ) {\displaystyle \mathrm {soc} (M)} is finitely generated and s o c ( M ) {\displaystyle \mathrm {soc} (M)} is an essential submodule of M {\displaystyle 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 r a d ( R ) {\displaystyle \mathrm {rad} (R)} , it is easy to see that r a d ( R ) {\displaystyle \mathrm {rad} (R)} annihilates s o c ( R ) {\displaystyle \mathrm {soc} (R)} . If R {\displaystyle R} is a finite-dimensional unital algebra and M {\displaystyle M} a finitely generated R {\displaystyle R} -module then the socle consists precisely of the elements annihilated by the Jacobson radical of R {\displaystyle 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.)
See also Injective hull Radical of a module Cosocle
References
Alperin, J.L.; Bell, Rowen B. (1995). Groups and Representations. Springer-Verlag. p. 136. ISBN 0-387-94526-1. Anderson, Frank Wylie; Fuller, Kent R. (1992). Rings and Categories of Modules. Springer-Verlag. ISBN 978-0-387-97845-1. Robinson, Derek J. S. (1996), A course in the theory of groups, Graduate Texts in Mathematics, vol. 80 (2 ed.), New York: Springer-Verlag, pp. xviii+499, doi:10.1007/978-1-4419-8594-1, ISBN 0-387-94461-3, MR 1357169