16.26 (2006)

We say that the prime graphs of finite groups $G$ and $H$ coincide if the sets of primes dividing their orders are the same, $\pi(G) = \pi(H)$, and for any distinct $p, q \in \pi(G)$ there is an element of order $pq$ in $G$ if and only if there is such an element in $H$. Does there exist a positive integer $k$ such that there are no $k$ pairwise non-isomorphic finite non-abelian simple groups with the same graphs of primes? Conjecture: $k = 5$.