We prove that every non-abelian finite simple group is generated by an involution and an element of prime order. Preliminary resultsThe p2, 3q-generation of classical groups has been studied extensively, and there are many results for groups of small dimension that we will make use of.Lemma 2.1. If G is a finite simple classical group listed in Table 2, then G is p2, 3qgenerated.We note that though there are many other p2, 3q-generation results regarding classical groups, our method will not require them.Lemma 2.2. If G is listed in Table 3 then G is p2, pq-generated, where p P t5, 7u is specified.Proof. Let G " P SL 3 p4q and p " 7. By [7] the only maximal subgroups M of G with order divisible by 7 are isomorphic to P SL 2 p7q. The index of M " P SL 2 p7q in G is 120

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Given non‐trivial finite groups A and B, not both of order 2, we prove that every finite simple group of sufficiently large rank is an image of the free product A*B. To show this, we prove that every finite simple group of sufficiently large rank is generated by a pair of subgroups isomorphic to A and B. This proves a conjecture of Tamburini and Wilson.

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