The main results of this paper are: Theorem A. Any finite nonabelian simple group can be generated by an involution and a Sylow 2-subgroup. - Theorem B. Let G be a finite group and K a field of characteristic p. If G acts faithfully on the irreducible KG-module V, then dim \(H^ 1(G,V)\leq (2/3)\dim V\). - Theorem C. Let G be a finite group of even order. Let O(G) be the maximal normal subgroup of G of odd order, and set \(\bar G=G/O(G)\). Then either G has a maximal subgroup of even index or \(A=O_ 2(\bar G)=\Phi (\bar G)\) and \(G/A\cong A_ 7\). - Theorem D. Let p be a prime and G a finite group. Then \(G=\) for some p-subgroup P and p'-subgroup R. The proofs of these results invoke the classification of finite simple groups. Various consequences of these results improve and extend work of several authors.