Khukhro, Evgeny (2012) Automorphisms of finite pgroups admitting a partition. Algebra and Logic, 51 (3). pp. 264277. ISSN 00025232
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Abstract
For a finite pgroup 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 ⋊ < φ>, where P 1 is a subgroup of index p and φ is a splitting automorphism of order p of P 1. It is proved that if a finite pgroup P with a partition admits a soluble group A of automorphisms of coprime order such that the fixedpoint 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 (Theorem 1). 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 pgroups with a partition. It is also shown that if a finite pgroup 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) (Theorem 2). The proof of this result has its basis in the author’s positive solution of an analog of the restricted Burnside problem for finite pgroups with a splitting automorphism of order p. Both theorems yield corollaries for finite groups admitting a Frobenius group of automorphisms whose kernel is generated by a splitting automorphism of prime order.
Additional Information:  • Translated from Algebra i Logika, Vol. 51, No. 3, pp. 392411, MayJune, 2012. 

Keywords:  Splitting automorphism, Finite pgroup, Exponent, Derived length, Nilpotency class, Frobenius group 
Subjects:  G Mathematical and Computer Sciences > G110 Pure Mathematics 
Divisions:  College of Science > School of Mathematics and Physics 
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ID Code:  15580 
Deposited On:  28 Oct 2014 11:31 
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