 
Summary: Generalized Conjugacy Classes
Pramod N. Achar
February 9, 1997
Abstract
Generalized conjugation is the action of a group on its underlying set given by (g, x) (g)xg1
,
where : G G is some fixed endomorphism. Here we study combinatorial properties of the sizes of the
orbits of the preceding action. In particular, we reduce the problem to a simpler case if has a nontrivial
kernel or if it is an inner automorphism, and we give a construction that allows a partial analysis in the
general case.
1 Introduction
1.1 Definition and Motivation
Let End G denote the semigroup of endomorphisms, and let SG denote the group of permutations on the
underlying set of G. Generalized conjugation associates to each element End G an action G SG of
G on its own uderlying set. In particular, if is the identity endomorphism, the associated action G SG
takes g G to the permutation which is conjugation by g. The computation of the associated element of
SG for given (, g) is described below.
Why do we wish to study generalized conjugation? Suppose that G is the fundamental group of some
compact 2manifold X, and that is induced by some continuous map f : X X. Consider the orbits in
G under the action associated to (referred to in the literature as "Reidemeister classes"): it turns out that
