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We determine the maximal hyperbolic reflection groups associated to the quadratic forms $-3x_0^2 + x_1^2 + ... + x_n^2$, $n ge 2$, and present the Coxeter schemes of their fundamental polyhedra. These groups exist in dimensions up to 13, and a proof is given that in higher dimensions these quadratic forms are not reflective.
Let $P(G)$ denotes the set of sizes of fibers of non-trivial commutators of the commutator word map. Here, we prove that $|P(G)|=1$, for any finite group $G$ of nilpotency class $3$ with exactlly two conjugacy class sizes. We also show that for given
Let $n geq 3$ be an integer. In this paper, we study the average behavior of the $2$-torsion in class groups of rings cut out by integral binary $n$-ic forms having any fixed odd leading coefficient. Specifically, we compute upper bounds on the avera
An abstract group $G$ is called totally 2-closed if $H = H^{(2),Omega}$ for any set $Omega$ with $Gcong Hleqtextrm{Sym}_Omega$, where $H^{(2),Omega}$ is the largest subgroup of symmetric group of $Omega$ whose orbits on $OmegatimesOmega$ are the same
We give a necessary and sufficient condition for a 2-dimensional or a three-generator Artin group $A$ to be (virtually) cocompactly cubulated, in terms of the defining graph of $A$.
In this paper, we study a group in which every 2-maximal subgroup is a Hall subgroup.