On the index of fixed point subgroup

Download

index.pdf

Date

2011

Author

Türkan, Erkan Murat

Metadata

Show full item record

Item Usage Stats

221
views

60
downloads

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 of G, we showed that when G is solvable, then the Fitting length f([G,A]) of [G,A] is bounded above by the number of primes (counted with multiplicities) dividing the index of fixed point subgroup of A in G and this bound is best possible.

Subject Keywords

Fixed point theory.

URI

http://etd.lib.metu.edu.tr/upload/12613522/index.pdf
https://hdl.handle.net/11511/21116

Collections

Graduate School of Natural and Applied Sciences, Thesis

Suggestions

OpenMETU
Core

Implications of the index of a fixed point subgroup Turkan, Erkan Murat (European Mathematical Society Publishing House, 2019-01-01) Let G be a finite group and A <= Aut(G). The index vertical bar G:C-G(A)vertical bar is called the index of A in G and is denoted by Ind(G)(A). In this paper, we study the influence of Ind(G)(A) on the structure of G and prove that [G, A] is solvable in case where A is cyclic, Ind G(A) is squarefree and the orders of G and A are coprime. Moreover, for arbitrary A <= Aut(G) whose order is coprime to the order of G, we show that when [G, A] is solvable, the Fitting height of [G, A] is bounded above by the num...
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.
Some sufficient conditions for p-nilpotency of a finite group Kızmaz, Muhammet Yasir (Informa UK Limited, 2019-09-02) Let G be a finite group and let p be prime dividing . In this article, we supply some sufficient conditions for G to be p-nilpotent (see Theorem 1.2) as an extension of the main theorem of Li et al. (J. Group Theor. 20(1): 185-192, 2017).
ON THE NUMBER OF QUADRATIC FORMS HAVING CODIMENSION 2 RADICALS IN CHARACTERISTIC 2 GIVING MAXIMAL/MINIMAL CURVES Özbudak, Ferruh (Informa UK Limited, 2014-09-02) Let F-q be an arbitrary finite field of characteristic 2 and k be an arbitrary even integer. We count the number of quadratic forms having codimension 2 radicals on F-q(k) over F-q such that the corresponding curve is maximal or minimal. This problem is first attempted in [3], in which the number of maximal curves is obtained only for (q, k) = (2, 6) and (q, k) = (2, 8).
Value sets of folding polynomials over finite fields Küçüksakallı, Ömer (2019-01-01) Let k be a positive integer that is relatively prime to the order of the Weyl group of a semisimple complex Lie algebra g. We find the cardinality of the value sets of the folding polynomials P-g(k)(x) is an element of Z[x] of arbitrary rank n >= 1, over finite fields. We achieve this by using a characterization of their fixed points in terms of exponential sums.

Citation Formats

E. M. Türkan, “On the index of fixed point subgroup,” Ph.D. - Doctoral Program, Middle East Technical University, 2011.