This article is cited in 6 papers

On the class of groups with pronormal Hall $\pi$-subgroups

W. Guo^a, D. O. Revin<sup>bc

a</sup> Department of Mathematics, University of Science and Technology of China, Hefei, P. R. China
^b Sobolev Institute of Mathematics, Novosibirsk, Russia
^c Novosibirsk State University, Novosibirsk, Russia

Abstract: Given a set $\pi$ of prime numbers, we define the class $\mathscr P_\pi$ of all finite groups in which Hall $\pi$-subgroups exist and are pronormal by analogy with the Hall classes $\mathscr E_\pi$, $\mathscr C_\pi$ and $\mathscr D_\pi$. We study whether $\mathscr P_\pi$ is closed under the main class-theoretic closure operations. In particular, we establish that $\mathscr P_\pi$ is a saturated formation.

Keywords: finite group, Hall subgroup, pronormal subgroup, class of finite groups, properties $\mathscr E_\pi$, $\mathscr C_\pi$ and $\mathscr D_\pi$, property $\mathscr P_\pi$, class-theoretic closure operations, formation, saturated formation, Fitting class.

UDC: 512.542.6

Received: 17.08.2013

 English version: Siberian Mathematical Journal, 2014, 55:3, 415–427

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