Related papers: Unitary t-groups

We show that in the extended modular group PGL(2,Z) there are exactly seven finite subgroups up to conjugacy; three subgroups of size 2, one subgroup each of size 3, 4, and 6, and the trivial subgroup of size 1.

It is proved that, for a prime $p>2$ and integer $n\geq 1$, finite $p$-groups of nilpotency class $3$ and having only two conjugacy class sizes $1$ and $p^n$ exist if and only if $n$ is even; moreover, for a given even positive integer,…

We construct all integral non-group-theoretical modular categories of dimension $p^2q^2$, where $p$ and $q$ are distinct prime numbers, establishing that a necessary and sufficient condition for their existence is that $p \mid q+1$, and…

It is well known that every finite simple group has a generating pair. Moreover, Guralnick and Kantor proved that every finite simple group has the stronger property, known as $\frac{3}{2}$-generation, that every nontrivial element is…

We prove that if the multiplication group $Mult(L)$ of a connected $2$-dimensional topological loop is a Lie group, then $Mult(L)$ is an elementary filiform nilpotent Lie group of dimension at least $4$. Moreover, we describe loops having…

We construct $(P_2)$-closed groups acting on $T_3$ in which all edge inversions have infinite order. This provides a negative answer to a question posed by Tornier. We also construct a family of $(P_2)$-closed groups for which the smallest…

We explicitly determine all CI-groups with respect to ternary relational structures that have the form $C \times D$, where $C$ is cyclic and $D$ is either a dicyclic group whose order is not divisible by $3$ or a dihedral group. Such groups…

The degree pattern of a finite group is the degree sequence of its prime graph in ascending order of vertices. We say that the problem of OD-characterization is solved for a finite group if we determine the number of pairwise nonisomorphic…

We obtain restrictions on units of even order in the integral group ring $\mathbb{Z}G$ of a finite group $G$ by studying their actions on the reductions modulo $4$ of lattices over the $2$-adic group ring $\mathbb{Z}_2G$. This improves the…

This note collects several results on the capability of $p$-groups of class two and prime exponent. Among the new results, we settle the 4-generator case for this class.

Let $n$ be a positive integer and $G(n)$ denote the number of non-isomorphic finite groups of order $n$. It is well-known that $G(n) = 1$ if and only if $(n,\phi(n)) = 1$, where $\phi(n)$ and $(a, b)$ denote the Euler's totient function and…

We study the existence of unitriangular basic sets for the symmetric group which behave nicely with respect to the Mullineux involution. Such sets give a natural labelling for the modular irreducible representations. We show that, for any…

An algebra is called a GI-algebra if its group of units satisfies a group identity. We provide positive support for the following two open problems. 1. Does every algebraic GI-algebra satisfy a polynomial identity? 2. Is every algebraically…

Let $G$ be a group. The permutability graph of subgroups of $G$, denoted by $\Gamma(G)$, is a graph having all the proper subgroups of $G$ as its vertices, and two subgroups are adjacent in $\Gamma(G)$ if and only if they permute. In this…

We prove that all hypergroups of order four are commutative and that there exists a non-comutative hypergroup of order five. These facts imply that the minimum order of non-commutative hypergroups is five even though the minimum order of…

In this paper we consider finite 2-groups with odd number of real conjugacy classes. On one hand we show that if $k$ is an odd natural number less than 24, then there are only finitely many finite 2-groups with exactly $k$ real conjugacy…

We consider the capability of $p$-groups of class two and odd prime exponent. The question of capability is shown to be equivalent to a statement about vector spaces and linear transformations, and using the equivalence we give proofs of…

Let U be the group of units of an infinite twisted group algebra K_\lambda G over a field K. We describe the maximal FC-subgroup of U and give a characterization of U with finitely conjugacy classes. In the case of group algebras we obtain…

In [5] the author conjectures and partially shows that the Cuntz semigroup classifies unitary elements of unital AF-algebras. We provide a complete proof by addressing the existence part of the conjecture, under a mild adjustment of both…

We determine the number of elements of order two in the group of normalized units V(F_2G) of the group algebra F_2G of a 2-group of maximal class over the field F_2 of two elements. As a consequence for the 2-groups G and H of maximal class…