# Kegel's theorem

## History

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

## Statement

### 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.

## 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.