that 2∈π3 is ruled out by showing that any such group must be a C-group and using Suzuki's classification of C-groups to check that none of the groups found by Suzuki satisfy this condition. and cover the cases when 2∈π4 and e≥3, or e=2. He shows that either G is a C-group so a Suzuki group, or satisfies his characterization of the groups E2(3) and S4(3) in his second paper, which are not N-groups. covers the case when 2∈π4 and e=1, where the only possibilities are that G is a C-group or the Tits group. Consequences A minimal simple group is a non-cyclic simple group all of whose proper subgroups are solvable. The complete list of minimal finite simple groups is given as follows PSL2(2p), p a prime. PSL2(3p), p an odd prime. PSL2(p), p > 3 a prime congruent to 2 or 3 mod 5 Sz(2p), p an odd prime. PSL3(3) In other words a non-cyclic finite simple group must have a subquotient isomorphic to one of these groups. References Finite groups