Related papers: Integer group determinants for small groups

Let $G$ be a finite group. In this short note, we give a criterion of nilpotency of $G$ based on the existence of elements of certain order in each section of $G$.

We classify all finite groups with five relative commutativity degrees. Also, we give a partial answer to our previous conjecture on a lower bound of the number of relative commutativity degrees of finite groups.

We study the problem of estimating the number of defective items in adaptive Group testing by using a minimum number of queries. We improve the existing algorithm and prove a lower bound that show that, for constant estimation, the number…

We introduce and study the class of spherically ordered groups. The notions of spherically ordered groups and their spectra of spherical orderability are introduced. Values of these spectra are found for a series of natural groups.

We calculate the precise number of r-regular elements in the finite exceptional groups. As a corollary we find that the proportion of r-regular elements is at least 3577/18432 and for all \epsilon>0, there are infinitely finite simple…

Using the computer algebra program GAP, we show that all crystallographic groups in dimensions at most 4 are distinguished from each other by their sets of finite quotients.

The main goal of this paper is to apply the arithmetic method developed in our previous paper \cite{13} to determine the number of some types of subgroups of finite abelian groups.

Let $G$ be a finite group and let $A_1,\ldots,A_k$ be a collection of subsets of $G$ such that $G=A_1\ldots A_k$ is the product of all the $A_i$'s with $|G|=|A_1|\ldots|A_k|$. We write $G=A_1\cdot\ldots\cdot A_k$ and call this a $k$-fold…

We evaluate determinants of "spiral" matrices, which are matrices in which entries are spiralling from the centre of the matrices towards the outside, with prescribed increments from one entry to the next depending on whether one moves…

We consider the computational problem of determining the unit group of a finite ring, by which we mean the computation of a finite presentation together with an algorithm to express units as words in the generators. We show that the problem…

We prove a general divisibility theorem that implies, e.g., that, in any group, the number of generating pairs (as well as triples, etc.) is a multiple of the order of the commutator subgroup. Another corollary says that, in any associative…

For every odd prime $p$ and every integer $n\geq 12$ there is a Heisenberg group of order $p^{5n/4+O(1)}$ that has $p^{n^2/24+O(n)}$ pairwise nonisomorphic quotients of order $p^{n}$. Yet, these quotients are virtually indistinguishable.…

We show that if a nontrivial group admits a locally invariant ordering, then it admits uncountably many locally invariant orderings. For the case of a left-orderable group, we provide an explicit construction of uncountable families of…

We provide a sufficient condition for a polynomial ring, not necessarily commutative, to have a first-order definition for the rational integers.

In this article we estimate the number of integers up to $X$ which can be represented by a positive-definite, binary integral quadratic form of discriminant which is small relative to $X$. This follows from understanding the vector of signs…

We associate with a matrix over an arbitrary field an infinite family of matrices whose sizes vary from one to infinity; their entries are traces of powers of the original matrix. We explicitly evaluate the determinants of matrices in our…

If V is a representation of a linear algebraic group G, a set S of G-invariant regular functions on V is called separating if the following holds: If two elements v,v' from V can be separated by an invariant function, then there is an f…

In this paper we study the ratio between the number of $p$-elements and the order of a Sylow $p$-subgroup of a finite group $G$. As well known, this ratio is a positive integer and we conjecture that, for every group $G$, it is at least the…

We consider models for social choice where voters rank a set of choices (or alternatives) by deliberating in small groups of size at most $k$, and these outcomes are aggregated by a social choice rule to find the winning alternative. We…

A group of matrices $G$ with entries in a number field $K$ is defined to be numerical if $G$ has a finite index subgroup of matrices whose entries are algebraic integers. It is shown that an irreducible or completely reducible subgroup of…