Related papers: The second maximal groups with respect to the sum …

Let C be an arbitrary simple-root cyclic code and let G be the subgroup of Aut(C) (the automorphism group of C) generated by the multiplier, the cyclic shift and the scalar multiplications. To the best of our knowledge, the subgroup G is…

Let $G$ be a finite group and $C_2$ the cyclic group of order 2. Consider the 8 multiplicative operations $(x,y)\mapsto (x^iy^j)^k$, where $i$, $j$, $k\in\{-1, 1\}$. Define a new multiplication on $G\times C_2$ by assigning one of the above…

A group $G$ is said to be equationally Noetherian if every system of equations in $G$ is equivalent to a finite subsystem. We show that all free-by-cyclic groups are equationally Noetherian. As a corollary, we deduce that the set of…

Let $G$ be a nonabelian group and $n$ a natural number. We say that $G$ has a strict $n$-split decomposition if it can be partitioned as the disjoint union of an abelian subgroup $A$ and $n$ nonempty subsets $B_1, B_2, \ldots, B_n$, such…

Let $G$ be a finite group, and let $r_{3}(G)$ represent the size of the largest subset of $G$ without non-trivial three-term progressions. In a recent breakthrough, Croot, Lev and Pach proved that $r_{3}(C_{4}^{n}) \leqslant (3.61)^{n}$,…

Let $p$ be a prime number. A longstanding conjecture asserts that every finite non-abelian $p$-group has a non-inner automorphism of order $p$. In this paper, we prove that if $G$ is an odd order finite non-abelian monolithic $p$-group such…

In this article, we study the metacyclic p-group codes arising from finite semisimple group algebras. In [CM25], we studied group codes arising from metacyclic groups with order divisible by two distinct odd primes. In the current work, we…

We determine the structure of the finite non-solvable groups of order divisible by $3$ all whose maximal subgroups of order divisible by $3$ are supersolvable. Precisely, we demonstrate that if $G$ is a finite non-solvable group satisfying…

Let $K$ be an algebraically closed field of characteristic $0$ and let $G$ be a finite cyclic group of order $n$. In this note we prove, using induction on the number of prime divisors of $n$, that $R_K(G)/I \cong \mathbb{Z}[X]/\langle…

In this note, we prove that $D_8\times C_2^{n-3}$ is the non-elementary abelian $2$-group of order $2^n$, $n\geq 3$, whose number of subgroups of possible orders is maximal. This solves a conjecture by Haipeng Qu [7]. A formula for counting…

Let $F$ be a $p$-adic field, that is, a finite extension of $\mathbb Q_p$. Let $D$ be a finite-dimensional central division algebra over $F$ and let $SL_1(D)$ be the group of elements of reduced norm $1$ in $D$. Prasad and Raghunathan…

A group is called $(m,n)$-bicyclic if it can be expressed as a product of two cyclic subgroups of orders $m$ and $n$, respectively. The classification and characterization of finite bicyclic groups have long been important problems in group…

Let $G$ be a finite group and $\Z G$ its integral group ring. We show that if $\alpha$ is a non-trivial bicyclic unit of $\Z G$, then there are bicyclic units $\beta$ and $\gamma$ of different types, such that $\GEN{\alpha,\beta}$ and…

We describe $\{2,3\}$-groups in which the order of a product of any two elements of orders at most $4$ does not exceed $9$ and the centralizer of every involution is a locally cyclic $2$-subgroup. In particular, we will prove that these…

A finite group of order divisible by 3 in which centralizers of 3-elements are 3-subgroups will be called a C{\theta}{\theta}-group. The prime graph (or Gruenberg-Kegel graph) of a finite group G is denoted by {\Gamma}(G) (or GK(G)) and its…

Let $Q$ be the matrix $\displaystyle \begin{pmatrix} a & b \\ 1 & 0 \end{pmatrix}$ in $GL_2(\mathbb{F}_q)$ where $\mathbb{F}_q$ is a finite field, and let $G$ be the finite cyclic group generated by $Q$. We consider the action of $G$ on the…

Let $D$ be a division ring with center $F$, and $G$ a subnormal subgroup of $D^*$. We show that if $G$ is a locally solvable group such that $G^{(i)}$ is algebraic over $F$, then $G$ must be central. Also, if $M$ is non-abelian locally…

Let $G$ be a finite group, written multiplicatively. The Davenport constant of $G$ is the smallest positive integer $D(G)$ such that every sequence of $G$ with $D(G)$ elements has a non-empty subsequence with product $1$. Let $D_{2n}$ be…

Given a finite group $G$, let $\pi(G)$ denote the set of all primes that divide the order of $G$. For a prime $r \in \pi(G)$, we define $r$-singular elements as those elements of $G$ whose order is divisible by $r$. Denote by $S_r(G)$ the…

Let $G$ be a finite abelian group $G$ with $N$ elements. In this paper we give a O(N) time algorithm for computing a basis of $G$. Furthermore, we obtain an algorithm for computing a basis from a generating system of $G$ with $M$ elements…