arXiv subjects

Istvan Kovacs

Publications and source records attributed to Istvan Kovacs.

3 recordsLinked to original sources

arXiv2023

Classification of cyclic groups underlying only smooth skew morphisms

A skew morphism of a finite group $A$ is a permutation $φ$ of $A$ fixing the identity element and for which there is an integer-valued function $π$ on $A$ such that $φ(ab)=φ(a)φ^{π(a)}(b)$ for all $a, b \in A$. A skew morphism $φ$ of $A$ is smooth if the associated power function $π$ is constant on the orbits of $φ$, that is, $π(φ(a))\equivπ(a)\pmod{|φ|}$ for all $a\in A$. In this paper we show that every skew morphism of a cyclic group of order $n$ is smooth if and only if $n=2^en_1$, where $0 \le e \le 4$ and $n_1$ is an odd square-free number. A partial solution to a similar problem on non-cyclic abelian groups is also given.

math.GR↗

arXiv2017

On groups all of whose Haar graphs are Cayley graphs

A Cayley graph of a group $H$ is a finite simple graph $Γ$ such that ${\rm Aut}(Γ)$ contains a subgroup isomorphic to $H$ acting regularly on $V(Γ)$, while a Haar graph of $H$ is a finite simple bipartite graph $Σ$ such that ${\rm Aut}(Σ)$ contains a subgroup isomorphic to $H$ acting semiregularly on $V(Σ)$ and the $H$-orbits are equal to the bipartite sets of $Σ$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that the groups $D_6, \, D_8, \, D_{10}$ and $Q_8$ are the only finite inner abelian groups all of whose Haar graphs are Cayley graphs (a group is called inner abelian if it is non-abelian, but all of its proper subgroups are abelian). As an application, it is also shown that every non-solvable group has a Haar graph which is not a Cayley graph.

math.CO↗

arXiv2014

Transitive Group Actions: (IM)PRIMITIVITY and Semiregular Subgroups

The following problem is considered: if $H$ is a semiregular abelian subgroup of a transitive permutation group $G$ acting on a finite set $X$, find conditions for (non) existence of $G$-invariant partitions of $X$. Conditions presented in this paper are derived by studying spectral properties of associated $G$-invariant digraphs. As an essential tool, irreducible complex characters of $H$ are used. Questions of this kind arise naturally when classifying combinatorial objects which enjoy a certain degree of symmetry. As an illustration, a new and short proof of an old result of Frucht, Graver and Watkins ({\it Proc. Camb. Phil. Soc.}, {\bf 70} (1971), 211-218) classifying edge-transitive generalized Petersen graphs, is given.

math.GR↗