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

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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.