On the orders of the non-Frattini elements of a finite group (Q1749373)

The prime graph of a finite group $G$ has the primes dividing $|G|$ as vertices and two vertices $p$, $q$ are joined when $pq$ divides the order of some element in $G$. The author considers the non-Frattini prime graph of $G$: it is the subgraph of the prime graph of $G$ defined by saying that two vertices $p$, $q$ are joined when $pq$ divides the order of some element of $G\setminus \varPhi (G)$. The main result, which is a particular case of a more technical one, asserts that the prime graph and the non-Frattini prime graph of $G$ coincide. \par The new (involving five technical conditions) notion of a $(p,q)$-Frattini-triple is introduced relative to a finite nonabelian simple group $G$ and a pair of primes dividing $|G|$ and it is used to obtain information on the orders of elements in $G\setminus \varPhi(G)$.