We show that the only finite nonabelian simple groups which admit a locally
linear, homologically trivial action on a closed simply connected 4-manifold
M (or on a 4-manifold with trivial first homology) are the alternating groups
A5, A6 and the linear fractional group PSL(2,7) (we note that for
homologically nontrivial actions all finite groups occur). The situation
depends strongly on the second Betti number b2(M) of M and has been known
before if b2(M) is different from two, so the main new result of the paper
concerns the case b2(M)=2. We prove that the only simple group that occurs
in this case is A5, and then give a short list of finite nonsolvable groups
which contains all candidates for actions of such groups.Comment: 17 page