The second stable homotopy group of the Eilenberg-Maclane space

We prove that for any group G, pi(S)(2) (K(G, 1)), the second stable homotopy group of the Eilenberg-Maclane space K(G, 1), is completely determined by the second homology group H-2(G, Z). We also prove that the second stable homotopy group pi(S)(2)(K(G, 1)) is isomorphic to H-2(G, Z)for a torsion group G with no elements of order 2 and show that for such groups, pi(S)(2)(K (G, 1)) is a direct factor of pi(3)(SK (G, 1)), where S denotes suspension and pi(S)(2) the second stable homotopy group. For radicable (divisible if G is abelian) groups G, we prove that pi(S)(2)(K(G, 1)) is isomorphic to H-2(G, Z). We compute pi(3)(SK (G, 1)) and pi(S)(2)(K (G, 1)) for symmetric, alternating, dihedral, general linear groups over finite fields and some infinite general linear groups G. For all finite groups G, we obtain a sharp bound for the cardinality of pi(S)(2)(K(G, 1)).