On the Hawkes graphs of finite groups

We study the properties and applications of the directed graph, introduced by Hawkes in 1968, of a finite group $G$. The vertex set of $\Gamma_H(G)$ coincides with $\pi(G)$ and $(p,q)$ is an edge if and only if $q\in \pi(G/O_{p',p}(G))$. In the language of properties of this graph we obtain commutation conditions for all $p$-elements with all $r$-elements of $G$, where $p$ and $r$ are distinct primes. We estimate the nilpotence length of a solvable finite group in terms of subgraphs of its Hawkes graph. Given an integer $n > 1$, we find conditions for reconstructing the Hawkes graph of a finite group $G$ from the Hawkes graphs of its $n$ pairwise nonconjugate maximal subgroups. Using these results, we obtain some new tests for the membership of a solvable finite group in the well-known saturated formations.