The Laitinen Conjecture for finite non-solvable groups
The Laitinen Conjecture for finite non-solvable groups
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DOI：
10.1017/s0013091512000223
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发表时间：
2012-12
影响因子：
0.7
通讯作者：
Krzysztof M. Pawałowski;Toshio Sumi
中科院分区：
文献类型：
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作者：
Krzysztof M. Pawałowski;Toshio Sumi
对任意有限群G，我们引入一个代数条件Gnil陪集条件，证明了满足Gnil陪集条件的任意有限奥利弗群G在具有孤立不动点的球面上有光滑作用，在孤立不动点处切G-模互不同构。此外，我们还证明了：对于任意不同构于Aut（A6）或P_n L（2，27）的有限不可解群G，Gnil陪集条件成立当且仅当rG ≥ 2，其中rG是G中非素幂阶元素的真实的共轭类的个数.作为结论，Laitinen猜想对任何不同构于Aut（A6）的有限非可解群成立。
Abstract For any finite group G, we impose an algebraic condition, the Gnil-coset condition, and prove that any finite Oliver group G satisfying the Gnil-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(A6) or PΣL(2, 27), the Gnil-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(A6).