On the Connectivity of the Sylow Graph of a Finite Group

The Sylow graph $\mathrm{\Gamma }(G)$ of a finite group $G$ originated from recent investigations on the so–called $\mathbf{N}$–closed classes of groups. The connectivity of $\mathrm{\Gamma }(G)$ was proved only few years ago, involving the classification of finite simple groups, and the structure of $G$ may be strongly restricted, once information on $\mathrm{\Gamma }(G)$ are given. The first result of the present paper deals with a condition on $\mathbf{N}$–closed classes of groups. The second result deals with a computational criterion, related to the connectivity of $\mathrm{\Gamma }(G)$.