Related papers: Integer group determinants for ${\rm C}_{4}^{2}$
We develop cycle index generating functions for orthogonal groups in even characteristic, and give some enumerative applications. A key step is the determination of the values of the complex linear-Weil characters of the finite symplectic…
In the following short paper we list some useful results concerning determinants and inverses of matrices. First we show, how to calculate determinants of $d \times d$ matrices, if their traces are known. As a next step $4 \times 4$…
We give an algorithm that decides whether a single equation in a group that is virtually a class $2$ nilpotent group with a virtually cyclic commutator subgroup, such as the Heisenberg group, admits a solution. This generalises the work of…
If $n>4$ and $c(\theta)$ denotes the set of irreducible constituents of a character $\theta$, then $c(\chi^k)={\rm Irr}(S_n)$ for all nonlinear $\chi\in {\rm Irr}(S_n)$ if and only if $k\geq n-1$.
Parametrization of complex 4 x 4 - matrices G in terms of Dirac tensor parameters (A,B,A_{l},B_{l},F_{kl}) or equivalent four complex 4-vectors (k,m,n,l) is investigated. In the given parametrization, the problem of inverting any 4 x 4…
We study the group of interval exchange transformations and obtain several characterizations of its commutator group. In particular, it turns out that the commutator group is generated by elements of order 2.
We give a formula for the determinant of an $n\times n$ matrix with entries from a commutative ring with unit. The formula can be evaluated by a "straight-line program" performing only additions, subtractions and multiplications of ring…
This contains a new version of the so-called non-commutative Gauss algorithm for polycyclic groups. Its results allow to read off the order and the index of a subgroup in an (possibly infinite) polycyclic group.
This paper investigates the critical group of a faithful representation of a finite group. It computes the order of the critical group in terms of the character values, and gives some restrictions on its subgroup structure. It also computes…
We compute the ring of non-induced representations for a cyclic group, $C_n$, over an arbitrary field and show that it has rank $\varphi(n)$, where $\varphi$ is Euler's totient function - independent of the characteristic of the field.…
In this paper, we present a novel approach for calculating the set of subgroups of a finite group, focusing on cyclic subgroups, and using it to establish the quantity of all subgroups in the direct product of two groups. Specifically, we…
One of the classical problems in group theory is determining the set of positive integers $n$ such that every group of order $n$ has a particular property $P$, such as cyclic or abelian. We first present the Sylow theorems and the idea of…
We define and we characterize regular and c-regular cyclically ordered abelian groups. We prove that every dense c-regular cyclically ordered abelian group is elementarily equivalent to some cyclically ordered group of unimodular complex…
An $integral$ of a group $G$ is a group $H$ whose commutator subgroup is isomorphic to $G$. This paper continues the investigation on integrals of groups started in the work arXiv:1803.10179. We study: (1) A sufficient condition for a bound…
In this article, we present an explicit formula for the $c$th nilpotent multiplier (the Baer invariant with respect to the variety of nilpotent groups of class at most $c\geq 1$) of the $n$th nilpotent product of some cyclic groups…
We introduce a coarse algebraic invariant for coarse groups and use it to differentiate various coarsifications of the group of integers. This lets us answer two questions posed by Leitner and the second author. The invariant is obtained by…
In this paper, first we obtain an explicit formula for an outer commutator multiplier of nilpotent products of cyclic groups with respect to the variety $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$, $\mathfrak{N}_{c}M(\mathbb{Z}\st{n}*…
In this paper, we consider the existence of group divisible designs (GDDs) with block size $4$ and group sizes $4$ and $7$. We show that there exists a 4-GDD of type $4^t 7^s$ for all but a finite specified set of feasible values for $(t,…
In this paper we introduce and study the concept of cyclic subgroup commutativity degree of a finite group $G$. This quantity measures the probability of two random cyclic subgroups of $G$ commuting. Explicit formulas are obtained for some…
Let $G$ be a finite group and let $c(G)$ be the number of cyclic subgroups of $G$. We study the function $\alpha(G) = c(G)/|G|$. We explore its basic properties and we point out a connection with the probability of commutation. For many…