Related papers: Enumeration of skew morphisms of cyclic $2$-groups

A skew morphism of a finite group $B$ is a permutation $\varphi$ of $B$ that preserves the identity element of $B$ and has the property that for every $a\in B$ there exists a positive integer $i_a$ such that $\varphi(ab) =…

A skew morphism of a finite group $A$ is a permutation $\varphi$ of $A$ fixing the identity element and for which there is an integer-valued function $\pi$ on $A$ such that $\varphi(ab)=\varphi(a)\varphi^{\pi(a)}(b)$ for all $a, b \in A$. A…

A skew morphism of a finite group $G$ is an element $\varphi$ of $\mathrm{Sym}(G)$ preserving the identity element of $G$ and having the property that for each $a\in G$ there exists a non-negative integer $i_a$ such that…

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…

A skew-morphism $\varphi$ of a finite group $A$ is a permutation on $A$ such that $\varphi(1)=1$ and $\varphi(xy)=\varphi(x)\varphi^{\pi(x)}(y)$ for all $x,y\in A$ where $\pi:A\to\mathbb{Z}_{|\varphi|}$ is an integer function. A…

A skew-morphism of a finite group $G$ is a permutation $\s$ on $G$ fixing the identity element, and for which there exists an integer function $\pi$ on $G$ such that $\s(xy)=\s(x)\s^{\pi(x)}(y)$ for all $x,y\in G$. It has been known that…

A skew-morphism of a finite group $G$ is a permutation $\sigma$ on $G$ fixing the identity element, and for which there exists an integer function $\pi$ on $G$ such that $\sigma(xy)=\sigma(x)\sigma^{\pi(x)}(y)$ for all $x,y\in G$. It has…

A cyclic complementary extension of a finite group $A$ is a finite group $G$ which contains $A$ and a cyclic subgroup $C$ such that $A\cap C=\{1_G\}$ and $G=AC$. For any fixed generator $c$ of the cyclic factor $C=\langle c\rangle$ of order…

A finite group $G$ is called a Schur group, if any Schur ring over $G$ is the transitivity module of a permutation group on the set $G$ containing the regular subgroup of all right translations. It was proved by R. P\"oschel (1974) that…

Skew morphisms, which generalise automorphisms for groups, provide a fundamental tool for the study of regular Cayley maps and, more generally, for finite groups with a complementary factorisation $X = GY$, where $Y$ is cyclic and core-free…

Skew morphisms, which generalise automorphisms for groups, provide a fundamental tool for the study of regular Cayley maps and, more generally, for finite groups with a complementary factorisation $G=BY$, where $Y$ is cyclic and core-free…

Rump proved in \cite[Theorem~1]{Rump2018ClassificationOC} that if a finite skew brace has cyclic additive group, then its multiplicative group is solvable and almost Sylow cyclic. In this paper we show that this rigidity persists when the…

An orthomorphism is a permutation $\sigma$ of $\{1, \dots, n-1\}$ for which $x + \sigma(x) \mod n$ is also a permutation on $\{1, \dots, n-1\}$. Eberhard, Manners, Mrazovi\'c, showed that the number of such orthomorphisms is $(\sqrt{e} +…

A dessin is a 2-cell embedding of a connected $2$-coloured bipartite graph into an orientable closed surface. A dessin is regular if its group of orientation- and colour-preserving automorphisms acts regularly on the edges. In this paper we…

A skew brace is a triplet $(A,\cdot,\circ)$, where $(A,\cdot)$ and $(A,\circ)$ are groups such that the brace relation $x\circ (y\cdot z) = (x\circ y)\cdot x^{-1}\cdot (x\circ z)$ holds for all $x,y,z\in A$. In this paper, we study the…

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 study the automorphism group of an infinite minimal shift $(X,\sigma)$ such that the complexity difference function, $p(n+1)-p(n)$, is bounded. We give some new bounds on $\mbox{Aut}(X,\sigma)/\langle \sigma \rangle$ and also study the…

A characterization is given of finite groups $H$ that have skew-morphisms of order coprime to the order $|H|$, and their skew-morphisms. A complete classification is then given of the automorphism groups and the underlying graphs of…

To a directed graph without loops and 2-cycles, we can associate a skew-symmetric matrix with integer entries. Mutations of such skew-symmetric matrices, and more generally skew-symmetrizable matrices, have been defined in the context of…

Let $G$ be a finite non-cyclic, non-characteristically simple group with the property that all proper characteristic subgroups of $G$ are cyclic. We call such a group $\mathrm{CCS}$ group, short for \emph{Characteristic Cyclic}. In this…