Related papers: Integer group determinants for ${\rm C}_{2}^{4}$

We prove that the theory of all modules over the ring of algebraic integers is decidable.

In this paper, we determine the 2-rank of the class group of certain classes of real cyclic quartic number fields. Precisely, we consider the case in which the quadratic subfield is Q(\sqrt{l}) with l=2 or a prime congruent to 1 mod 8.

This paper is devoted to the analysis of a false generalization of the rule of Sarrus and its properties that can be derived with the help of dihedral groups. Further, we discuss a Sarrus-like scheme that could be helpful for students to…

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…

The finite groups having an indecomposable polynomial invariant whose degree is at least half of the order of the group are classified. Apart from four sporadic exceptions these are exactly the groups having a cyclic subgroup of index at…

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 determine all groups definable in Presburger arithmetic, up to a finite index subgroup.

We enumerate the number of complex irreducible representations of each degree of general unitary groups of degree 4 over principal ideal rings of length 2.

In this paper we study some determinants and permanents. In particular, we investigate the new type determinants $$\det[(i^2+cij+dj^2)^{p-2}]_{1\le i,j\le p-1}\ \text{and} \ \det[(i^2+cij+dj^2)^{p-2}]_{0\le i,j\le p-1}$$ modulo an odd prime…

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 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.…

A group is called capable if it is a central factor group. We characterize the capable 2-generator 2-groups of class 2 in terms of a standard presentation.

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…

A standing conjecture in L2-cohomology is that every finite CW-complex X is of L2-determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class of groups containing e.g. all extensions of residually…

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…

Any Schur ring is uniquely determined by a partition of the elements of the group. An open question in the study of Schur rings is determining which partitions of the group induce a Schur ring. Although a structure theorem is available for…

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…

A ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive code ${\cal C}\subseteq{\mathbb{Z}}_2^\alpha\times{\mathbb{Z}}_4^\beta$ is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ${\mathbb{Z}}_2$ and the set of…

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…

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…