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Let $G$ be a finite group, and let $\text{Irr}(G)$ denote the set of the irreducible complex characters of $G$. An element $g\in G$ is called a vanishing element of $G$ if there exists $\chi\in\text{Irr}(G)$ such that $\chi(g)=0$ (i.e., $g$…

We construct a finitely generated group which is not virtually free, yet has decidable snake tiling problem. This shows that either a long-standing conjecture by Ballier and Stein (the characterization of groups with decidable domino…

In this note, we clarify that the boundary criterion for relative cubulation of the first author and Groves works even when the peripheral subgroups are not one-ended. Specifically, if the boundary criterion is satisfied for a relatively…

We show that the Lipschitz free space of a countable simplicial quasi-tree is isomorphic to $\ell^1$. As a consequence, every finitely generated group with Property (QT) of Bestvina--Bromberg--Fujiwara has a proper uniformly Lipschitz…

A cyclic subgroup graph of a group $G$ is a graph whose vertices are cyclic subgroups of $G$ and two distinct vertices $H_1$ and $H_2$ are adjacent if $H_1\leq H_2$, and there is no subgroup $K$ such that $H_1<K<H_2$. M.T\u{a}rn\u{a}uceanu…

We develop an algorithm for recognizing whether a character belongs to $\Sigma^m$. In order to apply it we just need to know that the ambient group is of type $\mathrm{FP}_m$ or of type $\mathrm{F}_2$ and that the word problem is solvable…

We call a finite group G ultrasolvable if it has a characteristic subgroup series whose factors are cyclic. It was shown by Durbin--McDonald that the automorphism group of an ultrasolvable group is supersolvable. The converse statement was…

The notion of Rota-Baxter groups was recently introduced by Guo, Lang and Sheng [{\em Adv. Math.} 387 (2021), 107834, 34 pp.] in the geometric study of Rota-Baxter Lie algebras. They are closely related to skew braces as observed by…

Let $\Gamma$ be a countable group and $\mathrm{Sub}(\Gamma)$ its Chabauty space, namely the compact $\Gamma$-space consisting of all subgroups of $\Gamma$. We call a subgroup $\Delta \in \mathrm{Sub}(\Gamma)$ a boomerang subgroup if for…

In this article, we give a description of the split exact sequences of left skew braces. We define a free action of the second cohomology group of a left skew brace $H$ by $Ann(I)$ on $Ext_{\alpha}(H, I)$ and show that this action becomes…

The class transposition group $CT(\mathbb{Z})$ was introduced by S. Kohl in 2010. It is a countable subgroup of the permutation group $Sym(\mathbb{Z})$ of the set of integers $\mathbb{Z}$. We study products of two class transpositions…

Reid-Smith recently parametrised groups acting on trees with Tits' independence property (P) using graph-based combinatorial structures known as local action diagrams. Properties of the acting (topological) group, such as being locally…

Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. Under some restrictions on the number of conjugacy classes of (non-normal) maximal subgroups of $G$, we prove that if $\sigma_1(G)<\frac{117}{20}\,$, then $G$ is…

We show that Plancherel convergence is strictly stronger than Benjamini-Schramm convergence. In order to do so, we introduce two criteria for Plancherel and Benjamini-Schramm convergence in terms of counting functions of the length…

Several algebraic criteria, reflecting displacement properties of transformation groups, have been used in the past years to prove vanishing of bounded cohomology and stable commutator length. Recently, the authors introduced the property…

We show that the Jordan decomposition of characters of finite reductive groups can be chosen so that if the centralizer of the relevant semisimple element in the dual group is connected, then the map is Galois-equivariant. Further, in this…

We show how to improve the runtime of the construction of generators of maximal subgroups of $\operatorname{SL}(d, q), \operatorname{SU}(d, q)$ and $\operatorname{Sp}(d, q)$ which arise as normalisers of extraspecial groups or $2$-groups of…

Let $N$ be a normal subgroup of a finite group $G$. From a result due to Brauer, it can be derived that the character table of $G$ contains square submatrices which are induced by the $G$-conjugacy classes of elements in $N$ and the…

The Liebeck-Nikolov-Shalev conjecture [LNS12] asserts that, for any finite simple non-abelian group $G$ and any set $A\subseteq G$ with $|A|\geq 2$, $G$ is the product of at most $N\frac{\log|G|}{\log|A|}$ conjugates of $A$, for some…

Given two subgroups $H,K$ of a compact group $G$, the probability that a random element of $H$ commutes with a random element of $K$ is denoted by $Pr(H,K)$. We show that if $G$ is a profinite group containing a Sylow $2$-subgroup $P$, a…