Related papers: Non-isomorphic metacyclic $p$-groups of split type…

Let $G$ be a finite group having a factorisation $G=AB$ into subgroups $A$ and $B$ with $B$ cyclic and $A\cap B=1,$ and let $b$ be a generator of $B$. The associated skew-morphism is the bijective mapping $f:A \to A$ well defined by the…

Let $X$ be a compact Riemann surface of genus $g\geq 2$, and let $G$ be a subgroup of $Aut(X)$. We show that if the Sylow $2$-subgroups of $G$ are cyclic, then $|G|\leq 30(g-1)$. If all Sylow subgroups of $G$ are cyclic, then, with two…

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…

We introduce a ``non-orientable'' variation of Serre's definition of a graph, which we call an abstract isogeny graph. These objects capture the combinatorics of the graphs $G(p,\ell,H)$, the $\ell$-isogeny graphs of supersingular elliptic…

Functional digraphs are unlabelled finite digraphs where each vertex has exactly one out-neighbor. They are isomorphic classes of finite discrete-time dynamical systems. Endowed with the direct sum and product, functional digraphs form a…

Let $G$ be a finite group. The group pseudo-algebra of $G$ is defined as the multi-set $C(G)=\{(d,m_G(d))\mid d\in{\rm Cod}(G)\},$ where $m_G(d)$ is the number of irreducible characters of with codegree $d\in {\rm Cod}(G)$. We show that…

The power graph $\mathcal{P}(G)$ of a group $G$ is the simple graph with vertex set $G$ and two vertices are adjacent whenever one of them is a positive power of the other. In this paper, for a finite noncyclic nilpotent group $G$, we study…

Finite $p$-groups of nilpotency class 2 are treated from the perspective of central extensions. Given finite abelian groups $G,A$, we derive an explicit formula for cocycles representing elements of $H^2(G,A)$, compute $H^2(G,A)$, and…

Let $G=C_{p^n}$ be a finite cyclic p-group, and let $Hol(G)$ denote its holomorph. In this work, we find and characterize the regular subgroups of $Hol(G)$ that are mutually normalizing each other in the permutation group $Sym(G)$. We…

Let $m,n,r$ be positive integers, and let $G=\langle a\rangle: \langle b\rangle \cong \mathbb{Z}_n: \mathbb{Z}_m$ be a split metacyclic group such that $b^{-1}ab=a^r$. We say that $G$ is {\em absolutely split with respect to $\langle…

Denote by $\omega(G)$ the number of orbits of the action of $Aut(G)$ on the finite group $G$. We prove that if $G$ is a finite nonsolvable group in which $\omega(G) \leqslant 5$, then $G$ is isomorphic to one of the groups…

Let $G$ be a finite group and $\alpha(G)=\frac{|C(G)|}{|G|}$\,, where $C(G)$ denotes the set of cyclic subgroups of $G$. In this short note, we prove that $\alpha(G)\leq\alpha(Z(G))$ and we describe the groups $G$ for which the equality…

The function $\mathrm{P}_{\mathbf{v}}(G)$, measuring the proportion of the elements of a finite group $G$ that are zeros of irreducible characters of $G$, takes (as proved in [12]) only values $\frac{m-1}{m}$, for $1 \leq m \leq 6$, in the…

For a finite non cyclic group $G$, let $\gamma(G)$ be the smallest integer $k$ such that $G$ contains $k$ proper subgroups $H_1,\dots,H_k$ with the property that every element of $G$ is contained in $H_i^g$ for some $i \in \{1,\dots,k\}$…

A group is called metahamiltonian if all non-abelian subgroups of it are normal. This concept is a natural generalization of Hamiltonian groups. In this paper, the properties of finite metahamiltonian $p$-groups are investigated.

The prime coprime graph $\Theta(G)$ of a finite group $G$ is the graph whose vertex set is $G$ and any two distinct vertices are adjacent if the greatest common divisor of their orders is either $1$ or a prime. In this paper, we investigate…

The number of subgroups and the number of cyclic subgroups are natural combinatorial invariants of a finite group. We investigate how restrictions on these quantities, together with the number of distinct prime divisors of $|G|$, enforce…

We study a class $\mathfrak{M}$ of cyclically presented groups that includes both finite and infinite groups and is defined by a certain combinatorial condition on the defining relations. This class includes many finite metacyclic…

Let $n>0$ be an integer and $\mathcal{X}$ be a class of groups. We say that a group $G$ satisfies the condition $(\mathcal{X},n)$ whenever in every subset with $n+1$ elements of $G$ there exist distinct elements $x,y$ such that $<x,y>$ is…

Let $G$ be a finite group, and let ${\rm{cd}}(G)$ denote the set of degrees of the irreducible complex characters of $G$. Define then the character degree graph $\Delta(G)$ as the (simple undirected) graph whose vertices are the prime…