JournalsrsmupVol. 136pp. 61–68

Automorphisms of finite order of nilpotent groups IV

  • B.A.F. Wehrfritz

    Queen Mary University of London, UK
Automorphisms of finite order of nilpotent groups IV cover
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Let ϕ\phi be an automorphism of finite order of the nilpotent group GG of class cc and mm and rr positive integers with ϕm=1\phi^{m} = 1. Consider the two (not usually homomorphic) maps ψ\psi and γ\gamma of GG given by

ψ ⁣:gggϕgϕ2gϕm1andγ ⁣:gg1gϕfor gG.\psi\colon g\longmapsto g\cdot g \phi \cdot g\phi^{2}\cdot\ldots\cdot g\phi^{m-1} \quad\text{and}\quad \gamma\colon g \mapsto g^{-1}\cdot g\phi\quad\text{for }g\in G.

We prove that the subgroups

X=xα ⁣:xkerψ,αAutG,xrs0(Gγ)s,X =\langle x\alpha\colon x\in\ker\:\psi, \alpha \in \mathrm {Aut}\: G, x^{r}\in\textstyle\bigcup_{s\geq 0}(G\gamma)^{s}\rangle,
Y=gγα ⁣:gG,αAutG,(gγ)rkerγ,Y =\langle g\gamma\alpha\colon g\in G, \alpha\in\mathrm {Aut}G\:, (g\gamma)^{r}\in \mathrm {ker}\:\gamma\rangle,
X=xrα ⁣:xkerψ,αAutG,xrs0(Gψ)s,X^{\ast} =\langle x^{r}\alpha\colon x\in\mathrm {ker}\:\psi, \in \alpha\in\mathrm{Aut} \:G, x^{r}\in\textstyle\bigcup_{s\geq 0}(G\psi)^{s} \rangle,
Y=(gγ)rα ⁣:gG,αAutG,(gγ)rkerγ=((Gγ)rkerγ)AutGY^{\ast}=\langle(g\gamma)^{r}\alpha\colon g\in G, \alpha\in\mathrm{Aut}\: G, (g\gamma)^{r}\in\mathrm {ker}\:\gamma\rangle=\langle((G\gamma)^{r}\cap\mathrm {ker}\:\gamma)\mathrm{Aut}\:G\rangle

of GG all have finite exponent bounded in terms of cc, mm and rr 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