The structure of groups in which every element has prime power order (CP-groups) is extensively studied. We first investigate the properties of group $G$ such that each element of $G\setminus N$ has prime power order. It is proved that $N$ is solvable or every non-solvable chief factor $H/K$ of $G$ satisfying $H\leq N$ is isomorphic to $PSL_2(3^f)$ with $f$ a 2-power. This partially answers the question proposed by Lewis in 2023, asking whether $G\cong M_{10}$? Furthermore, we prove that if each element $x\in G\backslash N$ has prime power order and ${\bf C}_G(x)$ is maximal in $G$, then $N$ is solvable. Relying on this, we give the structure of group $G$ with normal subgroup $N$ such that ${\bf C}_G(x)$ is maximal in $G$ for any element $x\in G\setminus N$. Finally, we investigate the structure of a normal subgroup $N$ when the centralizer ${\bf C}_G(x)$ is maximal in $G$ for any element $x\in N\setminus {\bf Z}(N)$, which is a generalization of results of Zhao, Chen, and Guo in 2020, investigating a special case that $N=G$ for our main result. We also provide a new proof for Zhao, Chen, and Guo's results above.

The Basilica group is a well-known 2-generated weakly branch, but not branch, group acting on the binary rooted tree. Recently, a more general form of the Basilica group has been investigated by Petschick and Rajeev, which is an $s$-generated weakly branch, but not branch, group that acts on the $m$-adic tree, for $s,m\ge 2$. A larger family of groups, which contains these generalised Basilica groups, is the family of iterated monodromy groups. With the new developments by Francoeur, the study of the existence of maximal subgroups of infinite index has been extended from branch groups to weakly branch groups. Here we show that a subfamily of iterated monodromy groups, which more closely resemble the generalised Basilica groups, have maximal subgroups only of finite index.

This paper studies the structure and preservational properties of lower bounded HNN extensions of inverse semigroups, as introduced by Jajcayová. We show that if $S^* = [ S;\; U_1,U_2;\; \phi ]$ is a lower bounded HNN extension then the maximal subgroups of $S^*$ may be described using Bass-Serre theory, as the fundamental groups of certain graphs of groups defined from the $\mathcal{D}$-classes of $S$, $U_1$ and $U_2$. We then obtain a number of results concerning when inverse semigroup properties are preserved under the HNN extension construction. The properties considered are completely semisimpleness, having finite $\mathcal{R}$-classes, residual finiteness, being $E$-unitary, and $0$-$E$-unitary. Examples are given, such as an HNN extension of a polycylic inverse monoid.

A Grigorchuk–Gupta–Sidki (GGS)-group is a subgroup of the automorphism group of the p-regular rooted tree for an odd prime p, generated by one rooted automorphism and one directed automorphism. Pervova proved that all torsion GGS-groups do not have maximal subgroups of infinite index. Here, we extend the result to nontorsion GGS-groups, which include the weakly regular branch, but not branch, GGS-group.

A subgroup $H$ is called a weak second maximal subgroup of $G$ if $H$ is a maximal subgroup of a maximal subgroup of $G$. Let $m(G,H)$ denote the number of maximal subgroups of $G$ containing $H$. We prove that $m(G,H)-1$ divides the index of some maximal subgroup of $G$ when $H$ is a weak second maximal subgroup of $G$. This partially answers a question of Flavell [‘Overgroups of second maximal subgroups’, Arch. Math.64(4) (1995), 277–282] and extends a result of Pálfy and Pudlák [‘Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups’, Algebra Universalis11(1) (1980), 22–27].

Let p ≥ 3 be a prime. A generalized multi-edge spinal group

$$G = \langle \{ a\} \cup \{ b_i^{(j)} {\rm \mid }1 \le j \le p,\, 1 \le i \le r_j\} \rangle \le {\rm Aut}(T)$$

$b^{(j)}_{1}, \ldots, b^{(j)}_{r_{j}}$

The work of L. G. (Laci) Kovács (1936–2013) gave us a deeper understanding of permutation groups, especially in the O’Nan–Scott theory of primitive groups. We review his contribution to this field.

Let $G$ be a finite group, let ${\it\pi}(G)$ be the set of prime divisors of $|G|$ and let ${\rm\Gamma}(G)$ be the prime graph of $G$. This graph has vertex set ${\it\pi}(G)$, and two vertices $r$ and $s$ are adjacent if and only if $G$ contains an element of order $rs$. Many properties of these graphs have been studied in recent years, with a particular focus on the prime graphs of finite simple groups. In this note, we determine the pairs $(G,H)$, where $G$ is simple and $H$ is a proper subgroup of $G$ such that ${\rm\Gamma}(G)={\rm\Gamma}(H)$.

We describe the current state of progress on the maximal subgroup problem for the Monster sporadic simple group. Any unknown maximal subgroup is an almost simple group whose socle is in one of 19 specified isomorphism classes.

2000 Mathematical Subject Classification:20D08.