ﻻ يوجد ملخص باللغة العربية
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. Poschel (1974) that given a prime $pge 5$ a $p$-group is Schur if and only if it is cyclic. We prove that a cyclic group of order $n$ is a Schur group if and only if $n$ belongs to one of the following five (partially overlapped) families of integers: $p^k$, $pq^k$, $2pq^k$, $pqr$, $2pqr$ where $p,q,r$ are distinct primes, and $kge 0$ is an integer.
A Cayley graph is said to be an NNN-graph if it is both normal and non-normal for isomorphic regular groups, and a group has the NNN-property if there exists an NNN-graph for it. In this paper we investigate the NNN-property of cyclic groups, and show that cyclic groups do not have the NNN-property.
A subset $B$ of an Abelian group $G$ is called a difference basis of $G$ if each element $gin G$ can be written as the difference $g=a-b$ of some elements $a,bin B$. The smallest cardinality $|B|$ of a difference basis $Bsubset G$ is called the diffe
Let $G$ be a finite cyclic group of order $n ge 2$. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)cdot ... cdot (n_lg)$ where $gin G$ and $n_1,..., n_l in [1,ord(g)]$, and the index $ind (S)$ of $S$ is defined as the minimum of $(n_
A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to a subset $S$ of $G$ is called normal if the right regular representation of $G$ is a normal subgroup in the full automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if for every
As a visualization of Cartier and Foatas partially commutative monoid theory, G.X. Viennot introduced heaps of pieces in 1986. These are essentially labeled posets satisfying a few additional properties. They naturally arise as models of reduced word