Rendiconti del Seminario Matematico della Università di Padova


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Volume 139, 2018, pp. 185–194
DOI: 10.4171/RSMUP/139-6

Published online: 2018-06-06

On an open problem about $\pi$-quasi-$\mathfrak{F}$-groups

Chi Zhang[1], Li Zhang[2] and Jianhong Huang[3]

(1) University of Science and Technology of China, Hefei, Anhui, China
(2) Anhui Jianzhu University, Hefei, Anhui, China
(3) Jiangsu Normal University, Xuzhou, Jiangsu, China

Let $\mathfrak{F}$ be a class of finite groups, $p$ a prime and $\pi$ a set of some primes. A finite group $G$ is called a $p$-quasi-$\mathfrak{F}$-group (respectively, by $\pi$-quasi-$\mathfrak{F}$-group) provided that for every $\mathfrak{F}$-eccentric $G$-chief factor $H/K$ of order divisible by $p$ (respectively, by at least one prime in $\pi$), the automorphisms of $H/K$ induced by all elements of $G$ are inner. In this paper, we obtain the characterizations of $p$-quasi-$\mathfrak{F}$-groups and $\pi$-quasi-$\mathfrak{F}$-groups, which give a positive answer to an open problem in the book [3].

Keywords: Finite groups, $p$-quasinilpotent group, fitting formation, $p$-quasi-$\mathfrak{F}$-group,\linebreak $\pi$-quasi-$\mathfrak{F}$-group

Zhang Chi, Zhang Li, Huang Jianhong: On an open problem about $\pi$-quasi-$\mathfrak{F}$-groups. Rend. Sem. Mat. Univ. Padova 139 (2018), 185-194. doi: 10.4171/RSMUP/139-6