We prove that for any group G, π2S(K(G,1))\documentclass[12pt]{minimal}
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\begin{document}$$\pi _2^S(K(G,1))$$\end{document}, the second stable homotopy group of the Eilenberg–Maclane space K(G, 1), is completely determined by the second homology group H2(G,Z)\documentclass[12pt]{minimal}
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\begin{document}$$H_2(G, \mathbb {Z})$$\end{document}. We also prove that the second stable homotopy group π2S(K(G,1))\documentclass[12pt]{minimal}
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\begin{document}$$\pi _2^S(K(G,1))$$\end{document} is isomorphic to H2(G,Z)\documentclass[12pt]{minimal}
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\begin{document}$$H_2(G, \mathbb {Z})$$\end{document} for a torsion group G with no elements of order 2 and show that for such groups, π2S(K(G,1))\documentclass[12pt]{minimal}
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\begin{document}$$\pi _2^S(K(G,1))$$\end{document} is a direct factor of π3(SK(G,1))\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{3}(SK(G,1))$$\end{document}, where S denotes suspension and π2S\documentclass[12pt]{minimal}
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\begin{document}$$\pi _2^S$$\end{document} the second stable homotopy group. For radicable (divisible if G is abelian) groups G, we prove that π2S(K(G,1))\documentclass[12pt]{minimal}
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\begin{document}$$\pi _2^S(K(G,1))$$\end{document} is isomorphic to H2(G,Z)\documentclass[12pt]{minimal}
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\begin{document}$$H_2(G, \mathbb {Z})$$\end{document}. We compute π3(SK(G,1))\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{3}(SK(G,1))$$\end{document} and π2S(K(G,1))\documentclass[12pt]{minimal}
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\begin{document}$$\pi _2^S(K(G,1))$$\end{document} 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))\documentclass[12pt]{minimal}
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\begin{document}$$\pi _2^S(K(G,1))$$\end{document}.
