Finite groups whose prime graphs are complete For a finite group $G$, the prime graph of $G$ is an undirected graph such that its vertices are all prime divisors of $\vert G\vert$ and two distinct vertices $p$ and $q$ are adjacent when there is an element in $G$ of order $pq$. 
Which finite groups does have a complete prime graph? i.e I am looking for finite groups in which for every two distinct prime divisors of $\vert G\vert$ say $p$ and $q$, there exists at least one element of order $pq$. For instance every Dedekind group has this property.
 A: To understand the "minimal troublemakers" in the case of solvable groups, I would start with a solvable group $G$ such that every proper section of $G$ has a complete prime graph but $G$ does not. Recall that a section of $G$ is a group $X/Y$ where $Y \lhd X$ and $X$ is  subgroup of $G$.
Such a group $G$ must be a $\{p,q\}$-group for a pair of distinct primes $p$ and $q$ (for otherwise $G$ has a Hall $\{p,q\}$-subgroup $H$ which is proper and $H$ contains an element of order $pq).$ Also, a Sylow $p$-subgroup of $G$ must be a maximal subgroup of $G$, and likewise a Sylow $q$-subgroup of $G$ must be maximal.
Now $G$ can't have both a normal Sylow $p$-subgroup and a normal Sylow $q$-subgroup. However, we have either $O_{p}(G) \neq 1$ or $O_{q}(G) \neq 1.$ Label so that $O_{p}(G) \neq 1.$ Then $G/O_{p}(G)$ must be a $q$-group (otherwise it contains an element of order $pq$, and then so does $G$).
Hence $O_{p}(G)$ is a normal Sylow $p$-subgroup of $G$, and is hence a maximal subgroup of $G$, so that $[G:O_{p}(G)] = q.$ Hence $G$ is a Frobenius group with kernel $O_{p}(G)$ and cyclic complement of order $q$.
We can go a little further since a Sylow $q$-subgroup $Q$ of $G$ is a maximal subgroup. For then $G = QM$ for some minimal normal subgroup of $G$ contained in $O_{p}(G),$ and we deduce that $|G|$ has the form $qp^{e}$ where $e$ is the smallest positive integer such that $q$ divides $p^{e}-1.$
A: This can be backed out from the referenced paper (here is the MathReview).
Maria Silvia Lucido and Ali Reza Moghaddamfar, MR 2063403 Groups with complete prime graph connected components, J. Group Theory 7 (2004), no. 3, 373--384.
