Automorphisms of finite order of nilpotent groups IV

B.A.F. Wehrfritz

doi:10.4171/rsmup/136-6

JournalsrsmupVol. 136pp. 61–68

Automorphisms of finite order of nilpotent groups IV

B.A.F. Wehrfritz
Queen Mary University of London, UK

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Abstract

Let $ϕ$ be an automorphism of finite order of the nilpotent group $G$ of class $c$ and $m$ and $r$ positive integers with $ϕ^{m} = 1$ . Consider the two (not usually homomorphic) maps $ψ$ and $γ$ of $G$ given by

ψ : g ⟼ g \cdot g ϕ \cdot g ϕ^{2} \cdot \dots \cdot g ϕ^{m - 1} and γ : g \mapsto g^{- 1} \cdot g ϕ for g \in G .

We prove that the subgroups

X = ⟨ xα : x \in ker ψ, α \in Aut G, x^{r} \in ⋃_{s \geq 0} (G γ)^{s} ⟩,

Y = ⟨ g γ α : g \in G, α \in Aut G, (g γ)^{r} \in ker γ ⟩,

X^{*} = ⟨ x^{r} α : x \in ker ψ, \in α \in Aut G, x^{r} \in ⋃_{s \geq 0} (G ψ)^{s} ⟩,

Y^{*} = ⟨(g γ)^{r} α : g \in G, α \in Aut G, (g γ)^{r} \in ker γ ⟩ = ⟨((G γ)^{r} \cap ker γ) Aut G ⟩

of $G$ all have finite exponent bounded in terms of $c$ , $m$ and $r$ only. This yields alternative proofs of the theorem of [4] and its related bounds.

Cite this article

B.A.F. Wehrfritz, Automorphisms of finite order of nilpotent groups IV. Rend. Sem. Mat. Univ. Padova 136 (2016), pp. 61–68

DOI 10.4171/RSMUP/136-6