On the influence of fixed point free nilpotent automorphism groups

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2017-12-01

Author

Ercan, Gülin

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A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that for all nonidentity elements . Let FH be a Frobenius-like group with complement H of prime order such that is of prime order. Suppose that FH acts on a finite group G by automorphisms where in such a way that In the present paper we prove that the Fitting series of coincides with the intersections of with the Fitting series of G, and the nilpotent length of G exceeds the nilpotent length of by at most one. As a corollary, we also prove that for any set of primes , the upper -series of coincides with the intersections of with the upper -series of G, and the - length of G exceeds the -length of by at most one.

Subject Keywords

Frobenius-like group, Fixed points, Nilpotent length, Pi-length

URI

https://hdl.handle.net/11511/44093

Journal

MONATSHEFTE FUR MATHEMATIK

DOI

https://doi.org/10.1007/s00605-016-0970-5

Collections

Department of Mathematics, Article

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Action of a Frobenius-like group with kernel having central derived subgroup Ercan, Gülin (2016-09-01) A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that [F, h] = F for all nonidentity elements h is an element of H. Suppose that a finite group G admits a Frobenius-like group of auto-morphisms FH of coprime order with [F', H] = 1. In case where C-G( F) = 1 we prove that the groups G and C-G( H) have the same nilpotent length under certain additional assumptions.
Derived length of a Frobenius-like kernel Ercan, Gülin; Khukhro, Evgeny (2014-08-15) 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 FH/[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 vertical bar FH vertical bar. It is proved that the derived length of the kernel F is bounded solely in terms of the dimension m = dim C-V(H...
Frobenius-like groups as groups of automorphisms Ercan, Gülin; Khukhro, Evgeny (2014-01-01) A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that FH/[F,F] is a Frobenius group with Frobenius kernel F/[F, F]. Such subgroups and sections are abundant in any nonnilpotent finite group. We discuss several recent results about the properties of a finite group G admitting a Frobenius-like group of automorphisms FH aiming at restrictions on G in terms of C-G(H) and focusing mainly on bounds for the Fitting height and rela...
Action of a Frobenius-like group Güloǧlu, Ismail Ş.; Ercan, Gülin (2014-03-15) We call a finite group Frobenius-like if it has a nontrivial nilpotent normal subgroup F possessing a nontrivial complement H such that [F, h] = F for all nonidentity elements h is an element of H. We prove that any irreducible nontrivial FH-module for a Frobenius-like group FH of odd order over an algebraically closed field has an H-regular direct summand if either F is fixed point free on V or F acts nontrivially on V and the characteristic of the field is coprime to the order of F. Some consequences of t...
NILPOTENT LENGTH OF A FINITE SOLVABLE GROUP WITH A FROBENIUS GROUP OF AUTOMORPHISMS Ercan, Gülin; Ogut, Elif (2014-01-01) We prove that a finite solvable group G admitting a Frobenius group FH of automorphisms of coprime order with kernel F and complement H such that [G, F] = G and C-CG(F) (h) = 1 for all nonidentity elements h is an element of H, is of nilpotent length equal to the nilpotent length of the subgroup of fixed points of H.

Citation Formats

G. Ercan, “On the influence of fixed point free nilpotent automorphism groups,” MONATSHEFTE FUR MATHEMATIK, pp. 531–538, 2017, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/44093.