Implications of the index of a fixed point subgroup
RENDICONTI DEL SEMINARIO MATEMATICO DELLA UNIVERSITA DI PADOVA, vol.142, pp.1-7, 2019 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 142
- Publication Date: 2019
- Doi Number: 10.4171/rsmup/26
- Journal Name: RENDICONTI DEL SEMINARIO MATEMATICO DELLA UNIVERSITA DI PADOVA
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
- Page Numbers: pp.1-7
- Open Archive Collection: AVESIS Open Access Collection
- Middle East Technical University Affiliated: No
Abstract
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 number of primes (counted with multiplicities) dividing Ind(G)(A) and this bound is best possible.