On the Fixed-Point Set and Commutator Subgroup of an Authomorphism of a Group of Finite Rank

Abstract

Let f be an automorphism of a group G. For G polycyclic, Endimioni and Moravec in [1] discuss the relationship between the fixed-point set C G (f) and the commutator subgroup [G, f] of f in G. Here we extend these results to soluble groups satisfying various rank restrictions. Suppose f is an automorphism of the polycyclic group G. Endimioni and Moravec in [1] prove the following. i) If C G (f) is finite and f has order 2, then [G, f] H is finite. ii) If C G (f) is finite, then so is G=G; f]. iii) If jfj is finite, then so is the index (G : [G, f].C G (f)). Of course C G (f) denotes the set of fixed points of f in G, [G, f] = h g À1 :gf : g P Gi and [G, f] is a f-invariant normal subgroup of G. Here we generalize these results to groups with some sort of rank restriction. Generally our groups at least have finite Hirsch number and satisfy something weaker than the FAR condition. We must start by defining these terms. If a group G has a series of finite length each factor of which is either locally finite or infinite cyclic, then the number of infinite cyclic factors in such a series is an invariant of G, which we call the Hirsch number of G (it is also sometimes called the torsion-free rank of G). This is the weakest of the rank restrictions considered here. The structure of groups with finite Hirsch number is discussed in many places (e.g. see [3]). From our point of view the most convenient description of the structure of such a group is

On the fixed-point set of an automorphism of a group

“…Theorem 4 strengthens Part ii) of the theorem of [9], which says that G/[G, φ] is finite. Throughout this paper if φ is an automorphism of a group G and m is a positive integer with φ m = 1, we define maps γ and ψ of G into itself by…”

“…These and related results are extended in [9] to, in particular, soluble groups of finite rank. Is there some stronger notion of [G, φ] being large such that in these situations C G (φ) is finite if and only is [G, φ] is large in this sense?…”

“…Then C Z (φ) = C ∩ Z = 1 and (Z : Zγ) is finite by Lemma 1. Since Zγ is infinite and torsion-free, the Hirsch number of G/Zγ is less than that of G. Further by Lemma 1c) of [10] the order of C G/Zγ (φ) divides (Zγ : (Zγ) m )|C|, which is a finite π-number. By induction there exists a normal subgroup N of G of finite index with Zγ ≤ N and N/Zγ lying in…”