# Kegel's theorem

From Groupprops

## Contents

## 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 and be two subgroups of a finite group , such that every conjugate of permutes with every conjugate of , and further, . Then one of and is contained in a proper normal subgroup of .

### 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

## Corollaries

- If 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 is a seminormal subgroup of and is a S-supplement of . Then by the theory of seminormal subgroups, every conjugate of is a S-supplement to . Hence, in particular, for any proper subgroup of , and are conjugate-permuting subgroups that do not generate the whole of . Hence, either of or is contained in a proper normal subgroup of .
- 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.