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