Finite groups having centralizer commutator product property

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.
RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO

Suggestions

Groups of automorphisms with TNI-centralizers
Ercan, Gülin (2018-03-15)
A subgroup H of a finite group G is called a TNI-subgroup if N-G(H) boolean AND H-9 = 1 for any g is an element of G \ N-G (H). Let A be a group acting on G by automorphisms where C-G(A) is a TNI-subgroup of G. We prove that G is solvable if and only if C-G(A) is solvable, and determine some bounds for the nilpotent length of G in terms of the nilpotent length of C-G(A) under some additional assumptions. We also study the action of a Frobenius group FH of automorphisms on a group G if the set of fixed point...
On the index of fixed point subgroup
Türkan, Erkan Murat; Ercan, Gülin; Department of Mathematics (2011)
Let G be a finite group and A be a subgroup of Aut(G). In this work, we studied the influence of the index of fixed point subgroup of A in G on the structure of G. When A is cyclic, we proved the following: (1) [G,A] is solvable if this index is squarefree and the orders of G and A are coprime. (2) G is solvable if the index of the centralizer of each x in H-G is squarefree where H denotes the semidirect product of G by A. Moreover, for an arbitrary subgroup A of Aut(G) whose order is coprime to the order o...
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.
On abelian group actions with TNI-centralizers
Ercan, Gülin (2019-07-03)
A subgroup H of a group G is said to be a TNI-subgroup if for any Let A be an abelian group acting coprimely on the finite group G by automorphisms in such a way that for all is a solvable TNI-subgroup of G. We prove that G is a solvable group with Fitting length h(G) is at most . In particular whenever is nonnormal. Here, h(G) is the Fitting length of G and is the number of primes dividing A counted with multiplicities.
A GENERALIZED FIXED-POINT-FREE ACTION
Güloğlu, İsmail Şuayip; Ercan, Gülin (2013-05-01)
In this paper we study the structure of a finite group G admitting a solvable group A of automorphisms of coprime order so that for any x epsilon C-G(A) of prime order or of order 4, every conjugate of x in G is also contained in C-G(A). Under this hypothesis it is proven that the subgroup [G, A] is solvable. Also an upper bound for the nilpotent height of [G, A] in terms of the number of primes dividing the order of A is obtained in the case where A is abelian.
Citation Formats
G. Ercan, “Finite groups having centralizer commutator product property,” RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO, pp. 341–346, 2015, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/35196.