JournalsrsmupVol. 142falsepp. 1–7

Implications of the index of a fixed point subgroup

  • Erkan Murat Türkan

    Çankaya University, Ankara, Turkey and Middle East Technical University, Ankara, Turkey
Implications of the index of a fixed point subgroup cover

A subscription is required to access this article.

Abstract

Let GG be a finite group and AAut(G)A\leq \operatorname{Aut}(G). The index G ⁣:CG(A)|G\colon C_G(A)| is called the index of A in G and is denoted by IndG(A)_G(A). In this paper, we study the influence of IndG(A)_G(A) on the structure of GG and prove that [G,A][G,{} A] is solvable in case where AA is cyclic, IndG(A)_G(A) is squarefree and the orders of GG and AA are coprime. Moreover, for arbitrary AAut(G)A\leq \operatorname{Aut}(G) whose order is coprime to the order of GG, we show that when [G,A][G,A] is solvable, the Fitting height of [G,A][G,A] is bounded above by the number of primes (counted with multiplicities) dividing IndG(A)_G(A) and this bound is best possible.

Cite this article

Erkan Murat Türkan, Implications of the index of a fixed point subgroup. Rend. Sem. Mat. Univ. Padova 142 (2019), pp. 1–7

DOI 10.4171/RSMUP/26