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This article is cited in 4 scientific papers (total in 4 papers)
Automorphisms of finite $p$-groups admitting a partition
E. I. Khukhro Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia
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 all elements outside which have order $p$; (c) to be a semidirect product $P=P_1\rtimes\langle\varphi\rangle$, where $P_1$ is a subgroup of index $p$ and $\varphi$ is its splitting automorphism of order $p$. It is proved that if a finite $p$-group $P$ with a partition admits a soluble automorphism group $A$ 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 derived by the author and P. V. 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 an automorphism group $A$ 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 has its basis in the author's positive solution of an analog of the restricted Burnside problem for finite $p$-groups with a splitting automorphism of order $p$. The results mentioned yield corollaries for finite groups admitting a Frobenius group of automorphisms whose kernel is generated by a splitting automorphism of prime order.
Keywords:
splitting automorphism, finite $p$-group, exponent, derived length, nilpotency class, Frobenius group of automorphisms.
Received: 29.01.2012 Revised: 24.03.2012
Citation:
E. I. Khukhro, “Automorphisms of finite $p$-groups admitting a partition”, Algebra Logika, 51:3 (2012), 392–411; Algebra and Logic, 51:3 (2012), 264–277
Linking options:
https://www.mathnet.ru/eng/al542 https://www.mathnet.ru/eng/al/v51/i3/p392
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Abstract page: | 295 | Full-text PDF : | 81 | References: | 46 | First page: | 6 |
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