The $Σ^1$ invariant for some Artin groups of arbitrary circuit rank

A finite group $G$ is called $k$-factorizable if for every ordered factorization $|G|=a_1\cdots a_k$ into integers each greater than $1$ there exist subsets $A_1,\dots,A_k\subseteq G$ such that $|A_i|=a_i$ for each $i$ and $G=A_1\cdots A_k$. The main results are as follows. 1. (Theorem 1.1) For every integer $k\geq3$ there exists a finite group $G$ such that $G$ is not $k$-factorizable. 2. (Theorem 1.2) Let $G$ be a finite group of order $4m$. If a Sylow $2$-subgroup of $G$ is elementary abelian and all involutions of $G$ are conjugate, then $G$ has no factorization of the form $G=ABC$ with $|A|=|C|=2$ and $|B|=m$. 3. Only $8$ groups of order at most $100$ fail to be $k$-factorizable for some $k$ (Theorem 1.3).

math.GR↗

arXiv2026

Corrigendum to "Transitive permutation groups with trivial four point stabilizers"

This article revisits earlier work by the last author together with Kay Magaard. We correct mistakes in results about the Sylow 2- and 3-structure of groups that act with fixity 3 and one result about possible components, we improve some results and we add details.

math.GR↗

arXiv2026

Spherical and affine Artin groups, finite quotients, and the isomorphism problem

We compute the smallest non-abelian quotients of spherical and affine Artin groups. Usually, these are isomorphic to the smallest non-abelian quotients of the corresponding Coxeter groups. As applications, we deduce that two affine Artin groups are isomorphic if and only if their Coxeter diagrams are isomorphic and that irreducible spherical Artin groups are distinguished from each other by their finite quotient groups.

math.GR↗