Finite permutation groups with large abelian quotients

L. G. Kovács and Cheryl Elisabeth Praeger

Vol. 136 (1989), No. 2, 283–292

DOI: 10.2140/pjm.1989.136.283

Abstract

We show that if G is a group of permutations on a set of n points and if |G∕G′| denotes the order of its largest abelian quotient, then either |G∕G′| = 1 or there is a prime p dividing |G∕G′| such that |G∕G′|≤ p^n∕p. Equality holds if and only if G is a p-group which is the direct product of its transitive constituents, with each of those having order p, except when p = 2 in which case one must also allow as transitive constituents the groups of order 4, the dihedral group of order 8 and degree 4, and the extraspecial group of order 32 and degree 8.

Mathematical Subject Classification 2000

Primary: 20B05

Milestones

Received: 13 November 1987

Published: 1 February 1989