## Abstract

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 H_{2}(G, Z). We also prove that the second stable homotopy group π2S(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, π2S(K(G,1)) is a direct factor of π_{3}(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 H_{2}(G, Z). We compute π_{3}(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)).

Original language | English |
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Pages (from-to) | 1327-1342 |

Number of pages | 16 |

Journal | Mathematische Zeitschrift |

Volume | 287 |

Issue number | 3-4 |

DOIs | |

State | Published - 1 Dec 2017 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2017, Springer-Verlag Berlin Heidelberg.

## Keywords

- Eilenberg–Maclane space
- Group actions
- Non-abelian tensor square
- Schur multiplier
- Second stable homotopy group

## ASJC Scopus subject areas

- General Mathematics