Derived length of a Frobenius-like kernel (original) (raw)

Journal of Algebra, 2014

Abstract

Abstract A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F called kernel which has a nontrivial complement H such that F H / [ F , F ] is a Frobenius group with Frobenius kernel F / [ F , F ] . Suppose that a Frobenius-like group FH acts faithfully by linear transformations on a vector space V over a field of characteristic that does not divide | F H | . It is proved that the derived length of the kernel F is bounded solely in terms of the dimension m = dim C V ( H ) of the fixed-point subspace of H by g ( m ) = 3 + [ log 2 ( m + 1 ) ] . It follows that if a Frobenius-like group FH acts faithfully by coprime automorphisms on a finite group G, then the derived length of the kernel F is at most g ( r ) , where r is the sectional rank of C G ( H ) . As an application, for a finite solvable group G admitting an automorphism φ of prime order coprime to | G | , a bound for the p-length of G is obtained in terms of the rank of a Hall p ′ -subgroup of C G ( φ ) . Earlier results of this kind were known only in the special case when the complement of the acting Frobenius-like group was assumed to have prime order and its fixed-point subspace (or subgroup) was assumed to be one-dimensional (or have all Sylow subgroups cyclic).

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