Minimal normal subgroup

A minimal normal subgroup is a nontrivial normal subgroup containing no smaller nontrivial normal subgroup of the ambient group. It need not be a minimal nontrivial subgroup. In a finite soluble group, it is elementary abelian: its derived subgroup must be trivial, a characteristic Sylow subgroup selects one prime, and the characteristic subgroup of elements killed by that prime gives exponent $p$.

A nontrivial minimal normal subgroup $N$ of a finite solvable group is abelian: its commutator subgroup is characteristic, and minimality plus solvability excludes $N^{\prime }=N$. A nontrivial Sylow subgroup of this abelian group is characteristic, so minimality makes $N$ a p-group. The characteristic subgroup of pth powers is proper, hence trivial by minimality. Therefore $N$ is an elementary abelian group.

Choose a minimal nontrivial normal subgroup $L$ of $K$. Its conjugates under the ambient group are minimal normal subgroups of $K$; distinct ones intersect trivially and commute. Choose a maximal family whose product is direct. Any omitted conjugate either lies in that product or meets it trivially and could be adjoined, so all conjugates lie in it. Their product is ambient-normal and therefore is $K$. A normal subgroup of one factor is then normal in $K$, since the other factors commute with it; minimality makes each factor simple. All factors are isomorphic because they are conjugates of $L$. In the abelian case they are cyclic of one common prime order.

The socle is generated by all minimal nontrivial normal subgroups. Distinct minimal normal subgroups commute and meet trivially, although in the abelian case not every collection of them is an independent direct-product decomposition. It is a direct product of simple groups. In a faithful primitive permutation group each nontrivial normal subgroup is transitive; its minimal normal subgroups are either elementary abelian and regular, or direct powers of a nonabelian simple group. This is the starting point of the O'Nan–Scott theorem.