The second stable homotopy group of the Eilenberg–Maclane space

We prove that for any group G, π2S(K(G,1)), the second stable homotopy group of the Eilenberg–Maclane space K(G, 1), is completely determined by the second homology group H₂(G, Z). We also prove that the second stable homotopy group π2S(K(G,1)) is isomorphic to H₂(G, Z) for a torsion group G with no elements of order 2 and show that for such groups, π2S(K(G,1)) is a direct factor of π₃(SK(G, 1)) , where S denotes suspension and π2S the second stable homotopy group. For radicable (divisible if G is abelian) groups G, we prove that π2S(K(G,1)) is isomorphic to H₂(G, Z). We compute π₃(SK(G, 1)) and π2S(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 π2S(K(G,1)).