A study of representations appeared as the tangential spaces of manifilds at fixed points

我研究了切空间在一个有限群作用的不动点在一个球。史密斯发表了一个问题是否切空间同构的行动领域正好有两个不动点。许多同行研究了相应的问题。我使用的技术由所罗门和Pawalowski和获得了一个充分条件有两个或两个以上的真实的共轭类的元素不是素数的权力秩序。我完全确定幂零群有两个或两个以上的真实的共轭类的元素不是素数的权力秩序。通过另一种结合上述方法的方法，我证明了在具有恰好两个不动点的球面上存在光滑作用，使得两个切向空间不同构，如果非素幂阶元素的真实的共轭类的数目大于1，对于有限的非-除了六字母交错群的自同构群之外的可解群和特殊线性射影群的域自同构扩张PSL（2，27）组。此外，我研究了可解群的情况，并表明该充分条件可以应用于某些类的可解群。

I studied about tangential spaces at the fixed point of a finite group action on a sphere. Smith posted a problem whether tangential spaces are isomorphic for an action on a sphere with exactly two fixed points. Many colleague studied the corresponding problem. I used the technique by Solomon and Pawalowski and obtained a sufficient condition to have two or more real conjugacy classes of elements not of prime power order. I completely determine nilpotent groups which have two or more real conjugacy classes of elements not of prime power order. By another approach combining the above I showed that there is a smooth action on a sphere with exactly two fixed points such that two tangencial spaces are not isomorphic, if the number of real conjugacy classes of elements not of prime power order is greater than 1, for a finite non-solvable groups except the automorphic group of the alternating group on six letters and the field automorphism extension for the projective special linear group PSL (2, 27). Furthermore, I studied the case of solvable groups and showed that the sufficient condition can be applied for some classes of solvable groups.