Finite groups admitting a dihedral group of automorphisms

Date

2017-01-01

Author

Ercan, Gülin

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Let D = alpha, beta be a dihedral group generated by the involutions alpha and beta and let F = alpha beta). Suppose that D acts on a finite group G by automorphisms in such a way that C-G(F)= 1. In the present paper we prove that the nilpotent, length of the group Cr' is equal to the maximum of the nilpotent lengths of the subgroups C-G (alpha) and C-G(beta).

Subject Keywords

Dihedral group, Fixed points, Nilpotent length

URI

https://hdl.handle.net/11511/55774

Journal

ALGEBRA & DISCRETE MATHEMATICS

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Department of Mathematics, Article

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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.
On the influence of fixed point free nilpotent automorphism groups Ercan, Gülin (2017-12-01) 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 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...
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.
A generalized fixed point free automorphism of prime power order Ercan, Gülin (2012-06-01) Let G be a finite group and alpha be an automorphism of G of order p(n) for an odd prime p. Suppose that alpha acts fixed point freely on every alpha-invariant p'-section of G, and acts trivially or exceptionally on every elementary abelian alpha-invariant p-section of G. It is proved that G is a solvable p-nilpotent group of nilpotent length at most n + 1, and this bound is best possible.
Finite groups having centralizer commutator product property Ercan, Gülin (2015-12-01) Let a be an automorphism of a finite group G and assume that G = {[g, alpha] : g is an element of G} . C-G(alpha). We prove that the order of the subgroup [G, alpha] is bounded above by n(log2(n+1)) where n is the index of C-G(alpha) in G.

Citation Formats

G. Ercan, “Finite groups admitting a dihedral group of automorphisms,” ALGEBRA & DISCRETE MATHEMATICS, pp. 223–229, 2017, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/55774.