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arXiv · 2505.20999

On a problem of B. Hartley about a small centralizer in finite and locally finite groups

Evgeny Khukhro
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Abstract

It is proved that if a finite group $G$ has an automorphism of order $n$ with $m$ fixed points, then $G$ has a soluble subgroup whose index and Fitting height are bounded in terms of $m$ and $n$. As a corollary, a problem of B. Hartley is solved in the affirmative: if a locally finite group $G$ has an element with finite centralizer, then $G$ has a subgroup of finite index which has a finite normal series with locally nilpotent factors.

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Evgeny Khukhro. 2025-08-20. On a problem of B. Hartley about a small centralizer in finite and locally finite groups. https://arxiv.org/abs/2505.20999

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