A complete description of all possible multiplicative groups of finite skew left braces whose additive group has trivial centre is given. As a consequence, some earlier results of Tsang can be improved and an answer to an open question set by Tsang at Ischia Group Theory 2024 Conference is provided.

We introduce the concept of almost $\mathcal {P}$-numbers where $\mathcal {P}$ is a class of groups. We survey the existing results in the literature for almost cyclic numbers, and give characterisations for almost abelian and almost nilpotent numbers proving these two concepts are equivalent.

Let $X=GC$ be a group, where C is a cyclic group and G is either a generalized quaternion group or a dihedral group such that $C\cap G=1$. In this paper, X is characterized and, moreover, a complete classification for $X$ is given, provided that G is a generalized quaternion group and C is core-free.

This note provides an affirmative answer to Problem 2.6 of Praeger and Schneider [‘Group factorisations, uniform automorphisms, and permutation groups of simple diagonal type’, Israel J. Math. 228(2) (2018), 1001–1023]. We will build groups $G$ (abelian, nonabelian and simple) for which there are two automorphisms $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}$ of $G$ such that the map

$$\begin{eqnarray}T=T_{\unicode[STIX]{x1D6FC}}\times T_{\unicode[STIX]{x1D6FD}}:G\longrightarrow G\times G,\quad g\mapsto (g^{-1}g^{\unicode[STIX]{x1D6FC}},g^{-1}g^{\,\unicode[STIX]{x1D6FD}})\end{eqnarray}$$

We derive some structural properties of a trifactorised finite group $G=AB=AC=BC$ , where $A$ , $B$ , and $C$ are subgroups of $G$ , provided that $A=A_{\unicode[STIX]{x1D70B}}\times A_{\unicode[STIX]{x1D70B}^{\prime }}$ and $B=B_{\unicode[STIX]{x1D70B}}\times B_{\unicode[STIX]{x1D70B}^{\prime }}$ are $\unicode[STIX]{x1D70B}$ -decomposable groups, for a set of primes $\unicode[STIX]{x1D70B}$ .

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.

Some classes of finitely generated hyperabelian groups defined in terms of semipermutability and S-semipermutability are studied in the paper. The classification of finitely generated hyperabelian groups all of whose finite quotients are PST-groups recently obtained by Robinson is behind our results. An alternative proof of such a classification is also included in the paper.

In this paper we analyse the structure of a finite group of minimal order among the finite non-supersoluble groups possessing a triple factorization by supersoluble subgroups of pairwise relatively prime indices. As an application we obtain some sufficient conditions for a triple factorized group by supersoluble subgroups of pairwise relatively prime indices to be supersoluble. Many results appear as consequences of our analysis.

The commutativity degree of a finite group is the probability that two randomly chosen group elements commute. The object of this paper is to compute the commutativity degree of a class of finite groups obtained by semidirect product of two finite abelian groups. As a byproduct of our result, we provide an affirmative answer to an open question posed by Lescot.

We find conditions which imply that a saturated fusion system over a product of 2-groups splits as a product of fusion systems over the factors.

Two subgroups A and B of a group G are said to be totally completely conditionally permutable (tcc-permutable) in G if X permutes with Yg for some g ∊ 〈X, Y〉, for all X ≤ A and Y ≤ B. We study the belonging of a finite product of tcc-permutable subgroups to a saturated formation of soluble groups containing all finite supersoluble groups.

Hypercentrally embedded subgroups of finite groups can be characterized in terms of permutability as those subgroups which permute with all pronormal subgroups of the group. Despite that, in general, hypercentrally embedded subgroups do not permute with the intersection of pronormal subgroups, in this paper we prove that they permute with certain relevant types of subgroups which can be described as intersections of pronormal subgroups. We prove that hypercentrally embedded subgroups permute with subgroups of prefrattini type, which are intersections of maximal subgroups, and with F-normalizers, for a saturated formation F. In the soluble universe, F-normalizers can be described as intersection of some pronormal subgroups of the group.

The authors describe the structure of the normal closure of a cyclic permutable subgroup of odd order in a finite group.