Replacement theorem

In mathematical group theory, the Thompson replacement theorem is a theorem about the existence of certain abelian subgroups of a p-group. The Glauberman replacement theorem is a generalization of it introduced by Glauberman (1968, Theorem 4.1).

Statement

Suppose that P is a finite p-group for some prime p, and let A be the set of abelian subgroups of P of maximal order. Suppose that B is some abelian subgroup of P. The Thompson replacement theorem says that if A is an element of A that normalizes B but is not normalized by B, then there is another element A* of A such that A*∩B is strictly larger than A∩B, and [A*,A] normalizes A.

The Glauberman replacement theorem is similar, except p is assumed to be odd and the condition that B is abelian is weakened to the condition that [B,B] commutes with B and with all elements of A. Glauberman says in his paper that he does not know whether the condition that p is odd is necessary.

References

• Glauberman, George (1968), "A characteristic subgroup of a p-stable group", Canadian Journal of Mathematics, 20: 1101–1135, doi:10.4153/cjm-1968-107-2, ISSN 0008-414X, MR 0230807
• Gorenstein, D. (1980), Finite Groups, New York: Chelsea, ISBN 978-0-8284-0301-6, MR 0569209
• Thompson, John G. (1969), "A replacement theorem for p-groups and a conjecture", Journal of Algebra, 13: 149–151, doi:10.1016/0021-8693(69)90068-4, ISSN 0021-8693, MR 0245683