Kegel's theorem

History
This result was stated and proved by O.H.Kegel in his work Produkte Nilpotenter Gruppen in ''Arch. Math.'' 1961.

Symbolic statement
Let $$A$$ and $$B$$ be two subgroups of a finite group $$G$$, such that every conjugate of $$A$$ permutes with every conjugate of $$B$$, and further, $$AB \ne G$$. Then one of $$A$$ and $$B$$ is contained in a proper normal subgroup of $$G$$.

Verbal statement
Given two conjugate-permuting subgroups that together do not generate the whole group, one of them is contained in a proper normal subgroup.

Related facts

 * Maximal conjugate-permutable implies normal
 * Conjugate-permutable implies subnormal in finite

Corollaries

 * If $$G$$ is a simple group, then the above situation (viz two proper subgroups whose every conjugate commutes but which don't generate the whole group) cannot occur.
 * Suppose $$A$$ is a seminormal subgroup of $$G$$ and $$B$$ is a S-supplement of $$A$$. Then by the theory of seminormal subgroups, every conjugate of $$B$$ is a S-supplement to $$A$$. Hence, in particular, for any proper subgroup $$C$$ of $$B$$, $$A$$ and $$C$$ are conjugate-permuting subgroups that do not generate the whole of $$G$$. Hence, either of $$A$$ or $$C$$ is contained in a proper normal subgroup of $$G$$.
 * Thus, in particular, in a simple group, the only possibility for a seminormal subgroup is one whose S-supplement is a cyclic group of prime order -- hence the group must be a subgroup of prime index.