Kegel's theorem

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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.

Related facts

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.