Narrow Results

A group is called HNN-free if it has no subgroups that are nontrivial HNN-extensions. We prove that finitely generated HNN-free implies virtually polycyclic for a large class of groups. We also consider finitely generated groups with no free subsemigroups of rank 2 and show that in many situations such groups are virtually nilpotent. Finally, as an application of our results, we determine the structure of locally graded groups in which every subgroup is pronormal, thus generalizing a theorem of Kuzennyi and Subbotin.

We introduce the concept of a conormal subgroup: a subgroup is conormal if it is contranormal in its normal closure. This unifies the concepts of normal and contranormal subgroups. We obtain some important properties of conormal subgroups, describe their connections with transitivity of normality, and study groups in which all conormal subgroups are normal.

A subgroup $H$ of a group $G$ is said to be pronormal in $G$ if $H$ and $H^g$ are conjugate in $〈H,H^g〉$ 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 ${\prod }_{i=1}^t{\mathbb{Z}}_3\wr {\mathrm{\text{Sym}}}_{n_i}$, where each ${\mathrm{\text{Sym}}}_{n_i}$ acts naturally on $\{1,\dots ,n_i\}$, such that $H$ projects onto ${\prod }_{i=1}^t{\mathrm{\text{Sym}}}_{n_i}$. Thus, in this paper, we obtain a criterion of pronormality of a subgroup $H$ of odd index in a subgroup of ${\prod }_{i=1}^t{\mathbb{Z}}_{p_i}\wr {\mathrm{\text{Sym}}}_{n_i}$, where each $p_i$ is a prime and each ${\mathrm{\text{Sym}}}_{n_i}$ acts naturally on $\{1,\dots ,n_i\}$, such that $H$ projects onto ${\prod }_{i=1}^t{\mathrm{\text{Sym}}}_{n_i}$.