Almost fixed-point-free automorphisms of soluble groups

Abstract

a b s t r a c tSuppose G is either a soluble (torsion-free)-by-finite group of finite rank or a soluble linear group over a finite extension field of the rational numbers. We consider the implications for G if G has an automorphism of finite order m with only finitely many fixed points. For example, if m is prime then G is a finite extension of a nilpotent group and if m = 4 then G is a finite extension of a centre-by-metabelian group. This extends the special cases where G is polycyclic, proved recently by Endimioni (2010); see [3].An automorphism φ of a group G is said to be almost fixed-point-free (a phrase we shorten to afpf) if its fixed-point set C G (φ) is finite. In [3] Endimioni proves that a polycyclic-by-finite group with an afpf automorphism of prime order is nilpotent-by-finite. Here we present a somewhat shorter alternative proof of this that also delivers the same conclusion for finite extensions of torsion-free soluble minimax groups and also for soluble-by-finite linear groups over finite extension fields of the rational numbers Q. Trivially polycyclic groups are (torsion-free)-by-finite, soluble, minimax and of finite rank and, less trivially (e.g. see [7], 1.2), (torsion-free)-by-finite soluble groups of finite rank have faithful finite-dimensional representations over the rationals Q. Neither reverse implication holds.If G is a finite soluble group with a fixed-point-free automorphism of prime order p, then G is nilpotent of class bounded by a function of p only-see Higman [4]; the least such function we denote here by h(p). (By a famous theorem of J.G. Thompson the solubility restriction is unnecessary, but we only require the soluble case.) Endimioni actually proves that if G is polycyclic-by-finite with an afpf automorphism of prime order p, then G has a nilpotent normal subgroup of finite index and class at most h(p). This also applies to our minimax, finite rank and linear cases. Below, Theorem 1 is our basic result and its proof delivers our proof of Endimioni's theorem. Theorem 2 we derive from Theorem 1 and Corollary 1 is immediate from Theorem 2 and [7], 1.2.

Almost fixed-point-free automorphisms of order 2

“…By Theorem 2, Corollary 1 of [10] the group G/D has an abelian normal subgroup of finite index (trivially so if φ does not have order 2 on G/D since then G/D is finite). By Lemma 1(c) we can choose this abelian normal subgroup to be inverted by φ.…”

“…If, however, we replace the word 'periodic' in the definition of finite Hirsch number by 'locally finite', then certainly something can be said. The torsion-free case goes through more-or-less as before, see [10], and would lead to G/T being nilpotent-by-finite, but the locally finite case is more problematic. See [4] and especially [4] Theorem 1.2 for a discussion of this.…”

“…The torsion-free case of Theorem 1 is covered by Theorem 2, Corollary 1 of [10]. The periodic case is also known; if T is a periodic group with an afpf automorphism of order 2, then by a theorem of Hartley and Meixner, see [5], T has a nilpotent subgroup of class at most 2 and of finite index in T .…”

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

“…But A p satisfies the minimal condition (G satisfies min-p for all primes p recall). Therefore A p A p g for all p and A Ag G; f. Also if K=A C G=A f, then Lemma 1 of [4] yields that K : A n. Now suppose the A T 1 is a divisible abelian p-group (the case s 1). Then Ag is divisible and hence is a direct summand of A.…”