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

Let $N(n)$ denote the number of isomorphism types of groups of order $n$. We consider the integers $n$ that are products of at most $4$ not necessarily distinct primes and exhibit formulas for $N(n)$ for such $n$.

An $integral$ of a group $G$ is a group $H$ whose derived group (commutator subgroup) is isomorphic to $G$. This paper discusses integrals of groups, and in particular questions about which groups have integrals and how big or small those…

The Profinite Isomorphism Problem for a class of groups \mathcal{C} asks for an algorithm that decides for any two groups in \mathcal{C} whether they have isomorphic profinite completions. We present the positive solution to this problem…

Based on the earlier work of Li (European J. Combin. 1997) and Dobson (Discrete Math. 2008), in this paper we complete the classification of cyclic $m$-DCI-groups and $m$-CI-groups. For a positive integer $m$ such that $m \ge 3$, we show…

Let $G$ be a group. A subset $D$ of $G$ is a determining set of $G$, if every automorphism of $G$ is uniquely determined by its action on $D$. The determining number of $G$, denoted by $\alpha(G)$, is the cardinality of a smallest…

Let $C_n$ be the cyclic group of order $n$. In this paper, we provide the exact values of some zero-sum constants over $C_n \rtimes_s C_2$ where $s \not\equiv \pm1 \pmod n$, namely $\eta$-constant, Gao constant, and Erd\H{o}s-Ginzburg-Ziv…

Let $p$ be a prime integer and $n,i$ be positive integers such that \linebreak $S=\{-1, \ \theta, \ \mu_i=1+\theta+... + \theta^{i-1} \ \mid 1 < i < \frac{p^n}{2}, \ gcd(p^n,i)=1 \}$ generates the group of units of $\mathbb{Z}[\theta],$…

We define a sequence of positive integers recursively, where each term is determined as follows: starting with a given positive integer, if the term is odd, the next is the sum of its positive divisors; if the term is even, the subsequent…

Let KG be a group algebra of a finite p-group G over a finite field K of characteristic p. We compute the order of the unitary subgroup of the group of units when G is either an extraspecial 2-group or the central product of such a group…

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…

Suppose $C(G)$ denotes the set of all cyclic subgroups of a finite group $G$, and $\mathcal{O}_{2}(G)$ denotes the number of elements of order $2$ in $G$. In [Marius T., Finite groups with a certain number of cyclic subgroups. The American…

We deduce a simple representation and the invariant factor decompositions of the subgroups of the group $\Bbb{Z}_m \times \Bbb{Z}_n$, where $m$ and $n$ are arbitrary positive integers. We obtain formulas for the total number of subgroups…

Taking the absolute value of consecutive differences of a cyclicly ordered list of integers constitutes a simple dynamical system. For lists of lenght a power of two the process will terminate in all zeros, but examples with arbitarily long…

In this paper, we consider linear combination of determinant and permanent, which we call generalized determinant, and determine the stabilizer group of it.

In this paper we explicitly determine all indicators for groups isomorphic to the semidirect product of two cyclic groups by an automorphism of prime order, as well as the generalized quaternion groups. We then compute the indicators for…

By the theory of elliptic curves, we study the integers representable as the product of the sum of four integers with the sum of their reciprocals and give a sufficient condition for the integers with a positive representation.

A description of the endomorphisms of semidirect products of two groups as a group of $2\times 2$ matrices of maps is already known. Using this description, we have studied the concept of determinant for the endomorphisms of semidirect…

Let $G$ be a permutation group of degree $n$ and let $s(G)$ denote the number of set-orbits of $G$. We determine $\inf(\frac {\log_2 s(G)} n)$ over all groups $G$ that satisfy certain restrictions on composition factors (i.e. $Alt(k), k >…

Let $C(n,p)$ be the set of $p$-compositions of an integer $n$, i.e., the set of $p$-tuples $\bm{\alpha}=(\alpha_1,...,\alpha_p)$ of nonnegative integers such that $\alpha_1+...+\alpha_p=n$, and $\mathbf{x}=(x_1,...,x_p)$ a vector of…

An integral of a group $G$ is a group $H$ whose commutator subgroup is isomorphic to $G$. In this paper, we prove that the integrability of a finite group is a decidable problem.