The socle \(Soc(G)\) is the product of all minimal normal subgroups. It gathers the irreducible bottom layer of normal structure: elementary abelian in solvable groups and possibly direct products of nonabelian simple groups in general. [@isaacs2008]
Definition
Minimal normality is relative to \(G\) and does not mean that the abstract group \(N\) is simple. A nonabelian minimal normal subgroup is usually a direct product of isomorphic simple groups.
Core result
\(M\cap N\) is normal in \(G\), so minimality makes it trivial or equal to each factor. In the trivial case, \([M,N]\leq M\cap N=1\). The simple-factor decomposition of a nonabelian minimal normal subgroup is developed in Isaacs, Section 9A.
Structural properties
- \(Soc(G)\operatorname{char}G\) because automorphisms permute the minimal normal subgroups.
- Minimal normal subgroups of a finite solvable group are elementary abelian, so \(Soc(G)\leq F(G)\).
- For every finite group, \(Soc(G)\leq F^*(G)\).
- Every nontrivial normal subgroup of a primitive permutation group is transitive; hence its socle is transitive, and an abelian socle is regular and elementary abelian.
- The socle of an almost simple group is its unique nonabelian simple normal subgroup.
The socle is not the product of all simple subgroups. It records only those appearing as minimal normal layers of the ambient group, so it is related to but distinct from the layer and Fitting subgroup.
Example and boundary
Knowledge network
The generalized Fitting subgroup contains the socle, normal closure constructs minimal normal layers, and primitive actions show how the socle determines permutation type.
Proof and sources
Direct-product behavior of minimal normal subgroups is in Section 2A; the abelian/semisimple split and relation to components are in Section 9A. [@isaacs2008]
