Transitive affine transformations on groups

David W. Jonah and Bertram Manuel Schreiber

Vol. 58 (1975), No. 2, 483–509

DOI: 10.2140/pjm.1975.58.483

Abstract

An affine transformation T on a group G is an automorphism followed by a translation; T is transitive if for each x,y ∈ G there is an integer n such that Tⁿ(x) = y. All groups with transitive affine transformations are determined: the infinite cyclic and infinite dihedral group are the only infinite examples; while the finite examples are semi-direct products of certain odd-order groups by a cyclic, dihedral or quaternion 2-group. The automorphism groups of the above groups are described, and the automorphisms which occur as parts of transitive affine transformations are given.

Mathematical Subject Classification

Primary: 20F15

Milestones

Received: 22 May 1974

Published: 1 June 1975