Normal p-complement

In mathematical group theory, a normal p-complement of a finite group for a prime p is a normal subgroup of order coprime to p and index a power of p. In other words the group is a semidirect product of the normal p-complement and any Sylow p-subgroup.

Frobenius normal p-complement theorem

The Frobenius normal p-complement theorem states that if the normalizer of every non-trivial subgroup of a Sylow p-subgroup of G has a normal p-complement, then so does G.

Glauberman normal p-complement theorem

The Glauberman normal p-complement theorem states that if p is an odd prime and the normalizer of Z(J(P)) has a normal p-complement, for P a Sylow p-subgroup of G, then so does G. Here Z stands for the center of a group and J for the Thompson subgroup.

The theorem fails for p=2: the simple group PSL₂(F₇) of order 168 is a counterexample.

References

• Gorenstein, D. (1980), Finite groups (2nd ed.), New York: Chelsea Publishing Co., ISBN 978-0-8284-0301-6, MR569209
• Thompson, John G. (1960), "Normal p-complements for finite groups", Mathematische Zeitschrift, 72: 332–354, doi:10.1007/BF01162958, ISSN 0025-5874, MR 0117289
• Thompson, John G. (1964), "Normal p-complements for finite groups", Journal of Algebra, 1: 43–46, doi:10.1016/0021-8693(64)90006-7, ISSN 0021-8693, MR0167521