Socle

Definition
The socle of a group can be defined as the subgroup generated by all minimal normal subgroups, i.e., the join of all minimal normal subgroups.

Facts

 * Socle of finite group is internal direct product of simple subnormal subgroups: Since minimal normal implies characteristically simple, the socle is generated by characteristically simple groups. In a finite group, a characteristically simple group is a direct product of simple groups, so the socle is generated by simple groups. It can further be shown that the socle is an internal direct product of a collection of simple subnormal subgroups.
 * Socle equals Omega-1 of center for nilpotent p-group: If $$G$$ is a group of prime power order, or more generally in a (possibly infinite) nilpotent p-group, the socle equals $$\Omega_1(Z(G))$$, i.e., the set of elements of prime order in the center of $$G$$ (along with the identity element). This follows from the fact that minimal normal implies central in nilpotent.
 * If $$G$$ is a finite solvable group, then the socle is a product of elementary abelian $$p$$-groups for a collection of primes dividing the order of $$G$$ (though this may not include all primes dividing the order of $$G$$). This follows from the fact that minimal normal implies elementary abelian in finite solvable. (When $$G$$ is nilpotent, all primes dividing its order are included).

Groups of prime power order
Here, the socle is Omega-1 of the center:

Finite solvable groups that are not nilpotent
Here, the socle is a product of elementary abelian groups for some of the primes dividing the order of the group:

Groups that are not solvable
Here, the socle is a product of simple groups, but we cannot say a priori whether it will comprise only simple abelian groups, simple non-abelian groups, or both.

Properties not satisfied
These are properties that are not always satisfied by the socle. They may be satisfied by the socle in a particular group.