We assign to each pair of positive integers $n$ and $k\ge 2$ a digraph $G(n,k)$ whose set of vertices is $H=\{0,1,\cdots ,n-1\}$ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^k\equiv b(modn)$. The digraph $G(n,k)$ is semiregular if there exists a positive integer $d$ such that each vertex of the digraph has indegree $d$ or 0. Generalizing earlier results of the authors for the case in which $k=2$, we characterize all semiregular digraphs $G(n,k)$ when $k\ge 2$ is arbitrary.

In this article, we study the elements with disconnected centralizer in the Brauer complex associated to a simple algebraic group $\mathbf{G}$ defined over a finite field with corresponding Frobenius map $F$ and derive the number of $F$-stable semisimple classes of $\mathbf{G}$ with disconnected centralizer when the order of the fundamental group has prime order. We also discuss extendibility of semisimple characters of the fixed point subgroup ${\mathbf{G}}^F$ to their inertia group in the full automorphism group. As a consequence, we...

Here we proved the existence of a closed vector space of sequences - any nonzero element of which does not comply with Schur’s property, that is, it is weakly convergent but not norm convergent. This allows us to find similar algebraic structures in some subsets of functions.

Let G be an additive finite abelian group. For every positive integer ℓ, let $dis{c}_{\ell }\left(G\right)$ be the smallest positive integer t such that each sequence S over G of length |S| ≥ t has a nonempty zero-sum subsequence of length not equal to ℓ. In this paper, we determine $dis{c}_{\ell }\left(G\right)$ for certain finite groups, including cyclic groups, the groups $G=C₂\oplus C_{2m}$ and elementary abelian 2-groups. Following Girard, we define disc(G) as the smallest positive integer t such that every sequence S over G with |S| ≥ t has nonempty zero-sum subsequences...

For a complex character $\chi$ of a finite group $G$, it is known that the product $\mathrm{sh}\left(\chi \right)={\prod }_{l\in L\left(\chi \right)}(\chi \left(1\right)-l)$ is a multiple of $\left|G\right|$, where $L\left(\chi \right)$ is the image of $\chi$ on $G-\left\{1\right\}$. The character $\chi$ is said to be a sharp character of type $L$ if $L=L\left(\chi \right)$ and $\mathrm{sh}\left(\chi \right)=\left|G\right|$. If the principal character of $G$ is not an irreducible constituent of $\chi$, then the character $\chi$ is called normalized. It is proposed as a problem by P. J. Cameron and M. Kiyota, to find finite groups $G$ with normalized sharp characters of type $\{-1,0,2\}$. Here we prove that such a group with nontrivial center is...

The paper reports the results of a search for pairs of groups of order $n$ that can be placed in the distance $n^2/4$ for the case when $n\in \{16,32\}$. The constructions that are used are of the general character and some of their properties are discussed as well.