Automorphisms of finite $p$-groups admitting a partition

Khukhro, E. I. (2012) Automorphisms of finite $p$-groups admitting a partition. [MIMS Preprint]

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Abstract

For a finite $p$-group $P$ the following three conditions are equivalent: (a) to have a (proper) partition, that is, to be the union of some proper subgroups with trivial pairwise intersections; (b) to have a proper subgroup outside of which all elements have order $p$; (c) to be a semidirect product $P=P_1\rtimes\langle \f\rangle$ where $P_1$ is a subgroup of index $p$ and $\f$ is a splitting automorphism of order $p$ of $P_1$. It is proved that if a finite $p$-group $P$ with a partition admits a soluble group of automorphisms $A$ of coprime order such that the fixed-point subgroup $C_P(A)$ is soluble of derived length $d$, then $P$ has a maximal subgroup that is nilpotent of class bounded in terms of $p$, $d$, and $|A|$. The proof is based on a similar result of the author and Shumyatsky for the case where $P$ has exponent $p$ and on the method of ``elimination of automorphisms by nilpotency'', which was earlier developed by the author, in particular, for studying finite $p$-groups with a partition. It is also proved that if a finite $p$-group $P$ with a partition admits a group of automorphisms $A$ that acts faithfully on $P/H_p(P)$, then the exponent of $P$ is bounded in terms of the exponent of $C_P(A)$. The proof of this result is based on the author's positive solution of the analogue of Restricted Burnside Problem for finite $p$-groups with a splitting automorphism of order $p$. Both theorems yield corollaries on finite groups admitting a Frobenius group of automorphisms whose kernel is generated by a splitting automorphism of prime order.

Item Type: MIMS Preprint
Uncontrolled Keywords: finite p-group with a partition; splitting automorphism; nilpotency class; autmorophism; fixed points
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 20 Group theory and generalizations
Depositing User: Professor Evgeny Khukhro
Date Deposited: 18 Oct 2012
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/1894

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