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For profinite branch groups, we first demonstrate the equivalence of the Bergman property, uncountable cofinality, Cayley boundedness, the countable index property, and the condition that every non-trivial normal subgroup is open; compact groups enjoying the last condition are called strongly just infinite. For strongly just infinite profinite branch groups with mild additional assumptions, we verify the invariant automatic continuity property and the locally compact automatic continuity property. Examples are then presented, including the profinite completion of the first Grigorchuk group. As an application, we show that many Burger–Mozes universal simple groups enjoy several automatic continuity properties.

Let G be a group of cardinality κ>ℵ0${\kappa>\aleph_{0}}$ endowed with a topology 𝒯${\mathcal{T}}$ such that |U|=κ${|U|=\kappa}$ for every non-empty U∈𝒯${U\in\mathcal{T}}$ and 𝒯${\mathcal{T}}$ has a base of cardinality κ. We prove that G can be factorized G=A⁢B${G=AB}$ (i.e. each g∈G${g\in G}$ has a unique representation g=a⁢b${g=ab}$, a∈A${a\in A}$, b∈B${b\in B}$) into dense subsets A , B , |A|=|B|=κ${|A|=|B|=\kappa}$. We do not know if this statement holds for κ=ℵ0${\kappa=\aleph_{0}}$ even if G is a topological group.

We define the concepts of projectively Krylov transitive and strongly projectively Krylov transitive Abelian p -groups. These concepts are non-trivial expansions of the projectively fully transitive and strongly projectively fully transitive Abelian p -groups, respectively, defined by the second author and Goldsmith in [13]. Some results concerning them and various other notions of transitivity are established as well, thus also generalizing achievements of Danchev–Goldsmith from [11]. Certain relationships between these classes are also obtained.

A PST-group is a group in which Sylow permutability is a transitive relation in the group. A classification is given of finitely generated hyperabelian groups all of whose finite quotients are PST-groups.

We classify the finite quasisimple groups whose commuting graph is perfect and we give a general structure theorem for finite groups whose commuting graph is perfect.

We provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element. We call an element of the Coxeter group parabolic quasi-Coxeter element if it has a factorization into a product of reflections that generate a parabolic subgroup. We give an unusual definition of a parabolic subgroup that we show to be equivalent to the classical one for finite Coxeter groups.

We address the question whether or not the condition on a fusion category being solvable is determined by its fusion rules. We prove that the answer is affirmative for some families of non-solvable examples arising from representations of semisimple Hopf algebras associated to exact factorizations of the symmetric and alternating groups. In the context of spherical fusion categories, we also consider the invariant provided by the S -matrix of the Drinfeld center and show that this invariant does determine the solvability of a fusion category provided it is group-theoretical.

Let σ={σi:i∈I}${\sigma=\{\sigma_{i}:i\in I\}}$ be some partition of the set of all primes ℙ${\mathbb{P}}$, G a finite group and σ⁢(G)={σi:σi∩π⁢(G)≠∅}${\sigma(G)=\{\sigma_{i}:\sigma_{i}\cap\pi(G)\neq\emptyset\}}$. A set 1∈ℋ${1\in{\mathcal{H}}}$ of subgroups of G is said to be a complete Hall σ-set of G if every non-identity group in ℋ${{\mathcal{H}}}$ is a Hall σi${\sigma_{i}}$-subgroup of G for some σi∈σ⁢(G)${\sigma_{i}\in\sigma(G)}$ and ℋ${{\mathcal{H}}}$ contains exactly one Hall σi${\sigma_{i}}$-subgroup of G for every σi∈σ⁢(G)${\sigma_{i}\in\sigma(G)}$. A subgroup H of G is called σ-permutable (resp. σ-permutably embedded ) in G if G possesses a complete Hall σ-set ℋ={1,H1,…,Ht}${{\mathcal{H}}=\{1,H_{1},\ldots,H_{t}\}}$ such that A⁢Hix=Hix⁢A${AH_{i}^{x}=H_{i}^{x}A}$ for any i and all x∈G${x\in G}$ (resp. if H has a complete Hall σ-set and every Hall σi${\sigma_{i}}$-subgroup of H is also a Hall σi${\sigma_{i}}$-subgroup of some σ-permutable subgroup of G ). In this paper, we classify the finite groups G such that either every maximal subgroup of every Sylow subgroup of G is σ-permutable in G or every maximal subgroup of every Sylow subgroup of G is σ-permutably embedded in G .

Let p be a prime number, let G be a finite group, and let P be a Sylow p -subgroup of G . Under the assumption that NG⁢(P)${N_{G}(P)}$ is p -nilpotent, we give an equivalent condition of the p -nilpotency of G . Our result is a generalization of the famous Burnside p -nilpotency criterion and some recent results.