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## 2012.94: Automorphisms of finite $p$-groups admitting a partition

2012.94: E. I. Khukhro (2012) Automorphisms of finite $p$-groups admitting a partition.

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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 finite p-group with a partition; splitting automorphism; nilpotency class; autmorophism; fixed points MSC 2000 > 20 Group theory and generalizations 2012.94 Professor Evgeny Khukhro 18 October 2012