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On the influence of fixed point free nilpotent automorphism groups
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Date
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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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.
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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...
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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
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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...
On the nilpotent length of a finite group with a frobenius group of automorphisms
Öğüt, Elif; Ercan, Gülin; Güloğlu, İsmail Ş.; Department of Mathematics (2013)
Let G be a finite group admitting a Frobenius group FH of automorphisms with kernel F and complement H. Assume that the order of G and FH are relatively prime and H acts regularly on the fixed point subgroup of F in G. It is proved in this thesis that the nilpotent length of G is less than or equal to the sum of the nilpotent length of the commutator group of G and F with 1 and the nilpotent length of the commutator group of G and F is equal to the nilpotent length of the fixed point subgroup of H in the co...
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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.