emph{The (undirected) power graph on the conjugacy classes} $mathcal{P_C}(G)$ of a group $G$ is a simple graph in which the vertices are the conjugacy classes of $G$ and two distinct vertices $C$ and $C'$ are adjacent in $mathcal{P_C}(G)$ if one is a subset of a power of the other. In this paper, we describe groups whose associated graphs are $k$-regular for $k=5,6$.

Uniformly describing the conjugacy classes of unipotent upper triangular groups \(\mathrm{UT}_{n}(\mathbb{F}_{q})\) (for all or many values \(n\) and \(q\)) is a nearly impossible task. This paper takes on related problem normal subgroups \(\mathrm{UT}_{n}(\mathbb{F}_{q})\). For \(q\) prime, bijection will be established between these pairs combinatorial objects with labels from \(\mathbb{F}_{q...

The tempered representations of a real reductive Lie group $G$ are naturally partitioned into series associated with conjugacy classes Cartan subgroups $H$ $G$. We incorporated partial Dirac cohomology for geometric construction various models these $H$--series, and show that this fits the framework quantization symplectic reduction.

fullerene chemistry is nowadays a well-established field of both theoretical and experimental investigations‎. this study considers the symmetry of small fullerenes cage c24 and c28‎. ‎using pm3 program for c24 and c28 fullerenes, oh and td symmetry were confirmed, respectively‎. ‎ the mentioned algorithm to compute the automorphism group of these fullerenes with connectivity and geometry of th...

in this paper we classify fuzzy subgroups of a rank-3 abelian group $g = mathbb{z}_{p^n} + mathbb{z}_p + mathbb{z}_p$ for any fixed prime $p$ and any positive integer $n$, using a natural equivalence relation given in cite{mur:01}. we present and prove explicit polynomial formulae for the number of (i) subgroups, (ii) maximal chains of subgroups, (iii) distinct fuzzy subgroups, (iv) non-isomorp...

let $g$ be a non-abelian group. the non-commuting graph $gamma_g$ of $g$ is defined as the graph whose vertex set is the non-central elements of $g$ and two vertices are joined if and only if they do not commute.in this paper we study some properties of $gamma_g$ and introduce $n$-regular $ac$-groups. also we then obtain a formula for szeged index of $gamma_g$ in terms of $n$, $|z(g)|$ and $|g|...