### Abstract

For any finite group G, we impose an algebraic condition, the G ^{nil}-coset condition, and prove that any finite Oliver group G satisfying the G ^{nil}-coset condition has a smooth action on some sphere with isolated fixed points at which the tangent G-modules are not isomorphic to each other. Moreover, we prove that, for any finite non-solvable group G not isomorphic to Aut(A 6) or PΣL(2, 27), the G ^{nil}-coset condition holds if and only if rG ≥ 2, where rG is the number of real conjugacy classes of elements of G not of prime power order. As a conclusion, the Laitinen Conjecture holds for any finite non-solvable group not isomorphic to Aut(A 6).

Original language | English |
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Pages (from-to) | 303-336 |

Number of pages | 34 |

Journal | Proceedings of the Edinburgh Mathematical Society |

Volume | 56 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 1 2013 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Proceedings of the Edinburgh Mathematical Society*,

*56*(1), 303-336. https://doi.org/10.1017/S0013091512000223

**The laitinen conjecture for finite non-solvable groups.** / Pawałowski, Krzysztof; Sumi, Toshio.

Research output: Contribution to journal › Article

*Proceedings of the Edinburgh Mathematical Society*, vol. 56, no. 1, pp. 303-336. https://doi.org/10.1017/S0013091512000223

}

TY - JOUR

T1 - The laitinen conjecture for finite non-solvable groups

AU - Pawałowski, Krzysztof

AU - Sumi, Toshio

PY - 2013/2/1

Y1 - 2013/2/1

N2 - For any finite group G, we impose an algebraic condition, the G nil-coset condition, and prove that any finite Oliver group G satisfying the G nil-coset condition has a smooth action on some sphere with isolated fixed points at which the tangent G-modules are not isomorphic to each other. Moreover, we prove that, for any finite non-solvable group G not isomorphic to Aut(A 6) or PΣL(2, 27), the G nil-coset condition holds if and only if rG ≥ 2, where rG is the number of real conjugacy classes of elements of G not of prime power order. As a conclusion, the Laitinen Conjecture holds for any finite non-solvable group not isomorphic to Aut(A 6).

AB - For any finite group G, we impose an algebraic condition, the G nil-coset condition, and prove that any finite Oliver group G satisfying the G nil-coset condition has a smooth action on some sphere with isolated fixed points at which the tangent G-modules are not isomorphic to each other. Moreover, we prove that, for any finite non-solvable group G not isomorphic to Aut(A 6) or PΣL(2, 27), the G nil-coset condition holds if and only if rG ≥ 2, where rG is the number of real conjugacy classes of elements of G not of prime power order. As a conclusion, the Laitinen Conjecture holds for any finite non-solvable group not isomorphic to Aut(A 6).

UR - http://www.scopus.com/inward/record.url?scp=84893043833&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84893043833&partnerID=8YFLogxK

U2 - 10.1017/S0013091512000223

DO - 10.1017/S0013091512000223

M3 - Article

AN - SCOPUS:84893043833

VL - 56

SP - 303

EP - 336

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

SN - 0013-0915

IS - 1

ER -