The second stable homotopy group of the Eilenberg–Maclane space

We prove that for any group G , π 2 S ( 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 π 2 S ( 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, π 2 S ( K ( G , 1 ) ) is a direct factor of π 3 ( S K ( G , 1 ) ) , where S denotes suspension and π 2 S the second stable homotopy group. For radicable (divisible if G is abelian) groups G , we prove that π 2 S ( K ( G , 1 ) ) is isomorphic to H 2 ( G , Z ) . We compute π 3 ( S K ( G , 1 ) ) and π 2 S ( 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 π 2 S ( K ( G , 1 ) ) .