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.^[1]

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 ${\displaystyle M}$ over a ring ${\displaystyle R}$ is defined to be the sum of the minimal nonzero submodules of ${\displaystyle M}$ . It can be considered as a dual notion to that of the radical of a module. In set notation,

${\displaystyle \mathrm {soc} (M)=\sum _{N{\text{ is a simple submodule of }}M}N.}$

Equivalently,

${\displaystyle \mathrm {soc} (M)=\bigcap _{E{\text{ is an essential submodule of }}M}E.}$

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

• If ${\displaystyle M}$ is an Artinian module, ${\displaystyle \mathrm {soc} (M)}$ is itself an essential submodule of ${\displaystyle M}$ .

In fact, if ${\displaystyle M}$ is a semiartinian module, then ${\displaystyle \mathrm {soc} (M)}$ is itself an essential submodule of ${\displaystyle M}$ . Additionally, if ${\displaystyle M}$ is a non-zero module over a left semi-Artinian ring, then ${\displaystyle \mathrm {soc} (M)}$ is itself an essential submodule of ${\displaystyle M}$ . 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 ${\displaystyle \mathrm {soc} (M)=M}$ . Rings for which ${\displaystyle \mathrm {soc} (M)=M}$ for all module ${\displaystyle M}$ are precisely semisimple rings.
• ${\displaystyle \mathrm {soc} (\mathrm {soc} (M))=\mathrm {soc} (M)}$ .
• ${\displaystyle M}$ is a finitely cogenerated module if and only if ${\displaystyle \mathrm {soc} (M)}$ is finitely generated and ${\displaystyle \mathrm {soc} (M)}$ is an essential submodule of ${\displaystyle M}$ .
• 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 ${\displaystyle \mathrm {rad} (R)}$ , it is easy to see that ${\displaystyle \mathrm {rad} (R)}$ annihilates ${\displaystyle \mathrm {soc} (R)}$ . If ${\displaystyle R}$ is a finite-dimensional unital algebra and ${\displaystyle M}$ a finitely generated ${\displaystyle R}$ -module then the socle consists precisely of the elements annihilated by the Jacobson radical of ${\displaystyle R}$ .^[2]

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.)^[3]

See also

• Injective hull
• Radical of a module
• Cosocle

References

1. Robinson 1996, p.87.
2. J. L. Alperin; Rowen B. Bell, Groups and Representations, 1995, ISBN 0-387-94526-1, p. 136
3. Mikhail Postnikov, Geometry VI: Riemannian Geometry, 2001, ISBN 3540411089,p. 98

• 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