On finite simple groups acting on integer and mod 2 homology 3-spheres

We prove that the only finite non-abelian simple groups G which possibly admit an action on a Z2-homology 3-sphere are the linear fractional groups PSL(2,q), for an odd prime power q (and the dodecahedral group A5≅PSL(2,5) in the case of an integer homology 3-sphere), by showing that G has dihedral Sylow 2-subgroups and applying the Gorenstein–Walter classification of such groups. We also discuss the minimal dimension of a homology sphere on which a linear fractional group PSL(2,q) acts.