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.

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