In 1960, Paul A. Smith asked the following question. If a finite group acts smoothly on a sphere with exactly two fixed points, is it true that the tangent –modules at the two points are always isomorphic? We focus on the case is an Oliver group and we present a classification of finite Oliver groups with Laitinen number or . Then we show that the Smith Isomorphism Question has a negative answer and for any finite Oliver group of odd order, and for any finite Oliver group with a cyclic quotient of order for two distinct odd primes and . We also show that with just one unknown case, this question has a negative answer for any finite nonsolvable gap group with . Moreover, we deduce that for a finite nonabelian simple group , the answer to the Smith Isomorphism Question is affirmative if and only if or .
"Smith equivalence and finite Oliver groups with Laitinen number 0 or 1." Algebr. Geom. Topol. 2 (2) 843 - 895, 2002. https://doi.org/10.2140/agt.2002.2.843