arXiv subjects
Publications and source records attributed to D. O. Revin.
This text is an extended version of the lecture notes for a course on representation theory of finite groups that was given by the authors during several years for graduate and postgraduate students of Novosibirsk State University and Sobolev Institute of Mathematics.
A subgroup $H$ of a group $G$ is said to be {\it pronormal} in $G$ if $H$ and $H^g$ are conjugate in $\langle H, H^g \rangle$ for each $g \in G$. Some problems in Finite Group Theory, Combinatorics, and Permutation Group Theory were solved in terms of pronormality, therefore, the question of pronormality of a given subgroup in a given group is of interest. Subgroups of odd index in finite groups satisfy a native necessary condition of pronormality. In this paper we continue investigations on pronormality of subgroups of odd index and consider the pronormality question for subgroups of odd index in some direct products of finite groups. In particular, in this paper we prove that the subgroups of odd index are pronormal in the direct product $G$ of finite simple symplectic groups over fields of odd characteristics if and only if the subgroups of odd index are pronormal in each direct factor of $G$. Moreover, deciding the pronormality of a given subgroup of odd index in the direct product of simple symplectic groups over fields of odd characteristics is reducible to deciding the pronormality of some subgroup $H$ of odd index in a subgroup of $\prod_{i=1}^t \mathbb{Z}_3\wr Sym_{n_i}$, where each $Sym_{n_i}$ acts naturally on $\{1,\dots, n_i\}$, such that $H$ projects onto $\prod_{i=1}^t Sym_{n_i}$. Thus, in this paper we obtain a criterion of pronormality of a subgroup $H$ of odd index in a subgroup of $\prod_{i=1}^t \mathbb{Z}_{p_i}\wr Sym_{n_i}$, where each $p_i$ is a prime and each $Sym_{n_i}$ acts naturally on $\{1,\dots, n_i\}$, such that $H$ projects onto $\prod_{i=1}^t Sym_{n_i}$.
Consider a triangle $ABC$ with given lengths $l_a,l_b,l_c$ of its internal angle bisectors. We prove that in general, it is impossible to construct a square of the same area as $ABC$ using a ruler and compass. Moreover, it is impossible to express the area of $ABC$ in radicals of $l_a,l_b,l_c$.
A subgroup $H$ of a group $G$ is said to be pronormal in $G$ if $H$ and $H^g$ are conjugate in $\langle H, H^g \rangle$ for every $g \in G$. In this paper we classify finite simple groups $E_6(q)$ and ${}^2E_6(q)$ in which all the subgroups of odd index are pronormal. Thus, we complete a classification of finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal.
A subgroup $H$ of a group $G$ is called {\it pronormal}, if for every $g\in G$ subgroups $H$ and $H^g$ are conjugate in $\langle H, H^g\rangle$. It is proven that if a finite group $G$ possesses a $π$-Hall subgroup for a set of primes $π$, the every its normal subgroup (in particular, $G$ itself) possesses a $π$-Hall subgroup that is pronormal in~$G$.
In this paper we find the number of conjugate $π$-Hall subgroups in all finite almost simple groups. We also complete the classification of $π$-Hall subgroups in finite simple groups and correct some mistakes from our previous paper.
A finite group $G$ is said to satisfy $C_π$ for a set of primes $π$, if $G$ possesses exactly one class of conjugate $π$-Hall subgroups. In the paper we obtain a criterion for a finite group $G$ to satisfy $C_π$ in terms of a normal series of the group.
In the paper we prove that the Hall property $C_π$ is inherited by normal subgroups which index is a $π'$-number.