On the realizability of a graph as the Gruenberg--Kegel graph of a finite group

N. V. Maslova ; D. Pagon

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 89-100

Résumé

Let $G$ be a finite group. Denote by $\pi(G)$ the set of all prime divisors of the order of $G$ and by $\omega (G)$ the spectrum of $G$, i.e. the set of all its element orders. The set $\omega(G)$ defines the Gruenberg–Kegel graph (or the prime graph) $\Gamma(G)$ of $G$; in this graph the vertex set is $\pi(G)$ and different vertices $p$ and $q$ are adjacent if and only if $pq\in\omega (G)$. We say that a graph $\Gamma$ with $|\pi(G)|$ vertices is realizable as the Gruenberg–Kegel graph of a group $G$ if there exists a vertices marking of $\Gamma$ by distinct primes from $\pi(G)$ such that the marked graph is equal to $\Gamma(G)$. A graph $\Gamma$ is realizable as the Gruenberg–Kegel graph of a group if $\Gamma$ is realizable as the Gruenberg–Kegel graph of an appropriate group $G$. We prove that a complete bipartite graph $K_{m,n}$ is realizable as the Gruenberg–Kegel graph of a group if and only if $m+n \le 6$ and $(m,n)\not =(3,3)$. Moreover, we describe all the groups $G$ such that the graph $K_{1,5}$ is realizable as the Gruenberg–Kegel graph of $G$.

Détail

Keywords: finite group, Gruenberg–Kegel graph (prime graph), realizability of a graph, complete bipartite graph.