Socle of a finite group ($\mathrm{soc}(G)=⟨N:N\text{ minimal normal in }G⟩$)

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.