Let G be a finite group. A subgroup A of G is said to be S-permutable in G if A permutes with every Sylow subgroup P of G, that is, $AP=PA$. Let $A_{sG}$ be the subgroup of A generated by all S-permutable subgroups of G contained in A and $A^{sG}$ be the intersection of all S-permutable subgroups of G containing A. We prove that if G is a soluble group, then S-permutability is a transitive relation in G if and only if the nilpotent residual $G^{\mathfrak {N}}$ of G avoids the pair $(A^{s G}, A_{sG})$, that is, $G^{\mathfrak {N}}\cap A^{sG}= G^{\mathfrak {N}}\cap A_{sG}$ for every subnormal subgroup A of G.

In this paper, we study triple-product-free sets, which are analogous to the widely studied concept of product-free sets. A nonempty subset S of a group G is triple-product-free if $abc \notin S$ for all $a, b, c \in S$. If S is triple-product-free and is not a proper subset of any other triple-product-free set, we say that S is locally maximal. We classify all groups containing a locally maximal triple-product-free set of size 1. We then derive necessary and sufficient conditions for a subset of a group to be locally maximal triple-product-free, and conclude with some observations and comparisons with the situation for standard product-free sets.

A subgroup A of a group G is said to be hereditarily G-permutable with a subgroup B of G, if $AB^x = B^xA$ for some element $x \in \langle A, B \rangle $. A subgroup A of a group G is said to be hereditarily G-permutable in G if A is hereditarily G-permutable with every subgroup of G. In this paper, we investigate the structure of a finite group G with all its Schmidt subgroups hereditarily G-permutable.

A subset R of the vertex set of a graph $\Gamma $ is said to be $(\kappa ,\tau )$-regular if R induces a $\kappa $-regular subgraph and every vertex outside R is adjacent to exactly $\tau $ vertices in R. In particular, if R is a $(\kappa ,\tau )$-regular set of some Cayley graph on a finite group G, then R is called a $(\kappa ,\tau )$-regular set of G. Let H be a nontrivial normal subgroup of G, and $\kappa $ and $\tau $ a pair of integers satisfying $0\leq \kappa \leq |H|-1$, $1\leq \tau \leq |H|$ and $\gcd (2,|H|-1)\mid \kappa $. It is proved that (i) if $\tau $ is even, then H is a $(\kappa ,\tau )$-regular set of G; (ii) if $\tau $ is odd, then H is a $(\kappa ,\tau )$-regular set of G if and only if it is a $(0,1)$-regular set of G.

Gagola and Lewis [‘A character theoretic condition characterizing nilpotent groups’, Comm. Algebra 27 (1999), 1053–1056] proved that a finite group G is nilpotent if and only if $\chi (1)^{2}$ divides $\lvert G:\textrm {ker}\,\chi \rvert $ for every irreducible character $\chi $ of G. The theorem was later generalised by using monolithic characters. We generalise the theorem further considering only strongly monolithic characters. We also give some criteria for solvability and nilpotency of finite groups by their strongly monolithic characters.

We describe finite soluble nonnilpotent groups in which every minimal nonnilpotent subgroup is abnormal. We also show that if G is a nonsoluble finite group in which every minimal nonnilpotent subgroup is abnormal, then G is quasisimple and $Z(G)$ is cyclic of order $|Z(G)|\in \{1, 2, 3, 4\}$ .

Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

Let G be a finite group and $\chi $ be a character of G. The codegree of $\chi $ is ${{\operatorname{codeg}}} (\chi ) ={|G: \ker \chi |}/{\chi (1)}$ . We write $\pi (G)$ for the set of prime divisors of $|G|$ , $\pi ({{\operatorname{codeg}}} (\chi ))$ for the set of prime divisors of ${{\operatorname{codeg}}} (\chi )$ and $\sigma ({{\operatorname{codeg}}} (G))= \max \{|\pi ({{\operatorname{codeg}}} (\chi ))| : \chi \in {\textrm {Irr}}(G)\}$ . We show that $|\pi (G)| \leq ({23}/{3}) \sigma ({{\operatorname{codeg}}} (G))$ . This improves the main result of Yang and Qian [‘The analog of Huppert’s conjecture on character codegrees’, J. Algebra478 (2017), 215–219].

For a given prime $p$, we investigate the finite groups all of whose 2-minimal $p$-subgroups are complemented.

Given a finite group $G$, define the minimal degree $\unicode[STIX]{x1D707}(G)$ of $G$ to be the least $n$ such that $G$ embeds into $S_{n}$. We call $G$ exceptional if there is some $N\unlhd G$ with $\unicode[STIX]{x1D707}(G/N)>\unicode[STIX]{x1D707}(G)$, in which case we call $N$ distinguished. We prove here that a subgroup with no abelian composition factors is not distinguished.

Let $g$ be an element of a finite group $G$ and let $R_{n}(g)$ be the subgroup generated by all the right Engel values $[g,_{n}x]$ over $x\in G$. In the case when $G$ is soluble we prove that if, for some $n$, the Fitting height of $R_{n}(g)$ is equal to $k$, then $g$ belongs to the $(k+1)$th Fitting subgroup $F_{k+1}(G)$. For nonsoluble $G$, it is proved that if, for some $n$, the generalized Fitting height of $R_{n}(g)$ is equal to $k$, then $g$ belongs to the generalized Fitting subgroup $F_{f(k,m)}^{\ast }(G)$ with $f(k,m)$ depending only on $k$ and $m$, where $|g|$ is the product of $m$ primes counting multiplicities. It is also proved that if, for some $n$, the nonsoluble length of $R_{n}(g)$ is equal to $k$, then $g$ belongs to a normal subgroup whose nonsoluble length is bounded in terms of $k$ and $m$. Earlier, similar generalizations of Baer’s theorem (which states that an Engel element of a finite group belongs to the Fitting subgroup) were obtained by the first two authors in terms of left Engel-type subgroups.

Let $\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${\mathcal{L}}_{\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}\in \mathfrak{F}$. A chief factor $H/K$ of $G$ is $\mathfrak{F}$-central in $G$ if $(H/K)\rtimes (G/C_{G}(H/K))\in \mathfrak{F}$. We study the structure of $G$ under the hypothesis that every chief factor of $G$ between $A_{G}$ and $A^{G}$ is $\mathfrak{F}$-central in $G$ for every subgroup $A\in {\mathcal{L}}_{\mathfrak{F}}(G)$. As an application, we prove that a finite soluble group $G$ is a PST-group if and only if $A^{G}/A_{G}\leq Z_{\infty }(G/A_{G})$ for every subgroup $A\in {\mathcal{L}}_{\mathfrak{N}}(G)$, where $\mathfrak{N}$ is the class of all nilpotent groups.

For every element $x$ of a finite group $G$, there always exists a unique minimal subnormal subgroup, say, $G_{x}$ of $G$ such that $x\in G_{x}$. The sub-class of $G$ in which $x$ lies is defined by $\{x^{g}\mid g\in G_{x}\}$. The aim of this paper is to investigate the influence of the sub-class sizes on the structure of finite groups.

Let $G$ be a finite group and $\unicode[STIX]{x1D70E}=\{\unicode[STIX]{x1D70E}_{i}\mid i\in I\}$ some partition of the set of all primes $\mathbb{P}$, that is, $\mathbb{P}=\bigcup _{i\in I}\unicode[STIX]{x1D70E}_{i}$ and $\unicode[STIX]{x1D70E}_{i}\cap \unicode[STIX]{x1D70E}_{j}=\emptyset$ for all $i\neq j$. We say that $G$ is $\unicode[STIX]{x1D70E}$-primary if $G$ is a $\unicode[STIX]{x1D70E}_{i}$-group for some $i$. A subgroup $A$ of $G$ is said to be: $\unicode[STIX]{x1D70E}$-subnormal in$G$ if there is a subgroup chain $A=A_{0}\leq A_{1}\leq \cdots \leq A_{n}=G$ such that either $A_{i-1}\unlhd A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is $\unicode[STIX]{x1D70E}$-primary for all $i=1,\ldots ,n$; modular in$G$ if the following conditions hold: (i) $\langle X,A\cap Z\rangle =\langle X,A\rangle \cap Z$ for all $X\leq G,Z\leq G$ such that $X\leq Z$ and (ii) $\langle A,Y\cap Z\rangle =\langle A,Y\rangle \cap Z$ for all $Y\leq G,Z\leq G$ such that $A\leq Z$; and $\unicode[STIX]{x1D70E}$-quasinormal in$G$ if $A$ is modular and $\unicode[STIX]{x1D70E}$-subnormal in $G$. We study $\unicode[STIX]{x1D70E}$-quasinormal subgroups of $G$. In particular, we prove that if a subgroup $H$ of $G$ is $\unicode[STIX]{x1D70E}$-quasinormal in $G$, then every chief factor $H/K$ of $G$ between $H^{G}$ and $H_{G}$ is $\unicode[STIX]{x1D70E}$-central in$G$, that is, the semidirect product $(H/K)\rtimes (G/C_{G}(H/K))$ is $\unicode[STIX]{x1D70E}$-primary.

Let $G$ be a group and $\unicode[STIX]{x1D70E}=\{\unicode[STIX]{x1D70E}_{i}\mid i\in I\}$ some partition of the set of all primes. A subgroup $A$ of $G$ is $\unicode[STIX]{x1D70E}$-subnormal in $G$ if there is a subgroup chain $A=A_{0}\leq A_{1}\leq \cdots \leq A_{m}=G$ such that either $A_{i-1}\unlhd A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is a finite $\unicode[STIX]{x1D70E}_{j}$-group for some $j=j(i)$ for $i=1,\ldots ,m$, and it is modular in $G$ if $\langle X,A\cap Z\rangle =\langle X,A\rangle \cap Z$ when $X\leq Z\leq G$ and $\langle A,Y\cap Z\rangle =\langle A,Y\rangle \cap Z$ when $Y\leq G$ and $A\leq Z\leq G$. The group $G$ is called $\unicode[STIX]{x1D70E}$-soluble if every chief factor $H/K$ of $G$ is a finite $\unicode[STIX]{x1D70E}_{i}$-group for some $i=i(H/K)$. In this paper, we describe finite $\unicode[STIX]{x1D70E}$-soluble groups in which every $\unicode[STIX]{x1D70E}$-subnormal subgroup is modular.

Kang and Liu [‘On supersolvability of factorized finite groups’, Bull. Math. Sci.3 (2013), 205–210] investigate the structure of finite groups that are products of two supersoluble groups. The goal of this note is to give a correct proof of their main theorem.

Let $\unicode[STIX]{x1D70E}=\{\unicode[STIX]{x1D70E}_{i}\mid i\in I\}$ be a partition of the set of all primes $\mathbb{P}$. Let $\unicode[STIX]{x1D70E}_{0}\in \unicode[STIX]{x1D6F1}\subseteq \unicode[STIX]{x1D70E}$ and let $\mathfrak{I}$ be a class of finite $\unicode[STIX]{x1D70E}_{0}$-groups which is closed under extensions, epimorphic images and subgroups. We say that a finite group $G$ is $\unicode[STIX]{x1D6F1}_{\mathfrak{I}}$-primary provided $G$ is either an $\mathfrak{I}$-group or a $\unicode[STIX]{x1D70E}_{i}$-group for some $\unicode[STIX]{x1D70E}_{i}\in \unicode[STIX]{x1D6F1}\setminus \{\unicode[STIX]{x1D70E}_{0}\}$ and we say that a subgroup $A$ of an arbitrary group $G^{\ast }$ is $\unicode[STIX]{x1D6F1}_{\mathfrak{I}}$-subnormal in $G^{\ast }$ if there is a subgroup chain $A=A_{0}\leq A_{1}\leq \cdots \leq A_{t}=G^{\ast }$ such that either $A_{i-1}\unlhd A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is $\unicode[STIX]{x1D6F1}_{\mathfrak{I}}$-primary for all $i=1,\ldots ,t$. We prove that the set ${\mathcal{L}}_{\unicode[STIX]{x1D6F1}_{\mathfrak{I}}}(G)$ of all $\unicode[STIX]{x1D6F1}_{\mathfrak{I}}$-subnormal subgroups of $G$ forms a sublattice of the lattice of all subgroups of $G$ and we describe the conditions under which the lattice ${\mathcal{L}}_{\unicode[STIX]{x1D6F1}_{\mathfrak{I}}}(G)$ is modular.

Let $G$ be a finite group and let $A\left( G \right)$ denote the Burnside ring of $G$. Then an inverse limit $L\left( G \right)$ of the groups $A\left( H \right)$ for proper subgroups $H$ of $G$ and a homomorphism res from $A\left( G \right)$ to $L\left( G \right)$ are obtained in a natural way. Let $Q\left( G \right)$ denote the cokernel of res. For a prime $p$, let $N\left( p \right)$ be the minimal normal subgroup of $G$ such that the order of ${G}/{N}\;\left( p \right)$ is a power of $p$, possibly 1. In this paper we prove that $Q\left( G \right)$ is isomorphic to the cartesian product of the groups $Q\left( {G}/{N\left( p \right)}\; \right)$, where $p$ ranges over the primes dividing the order of $G$.