Smith equivalence and finite Oliver groups with Laitinen number 0 or 1

Abstract

In 1960, Paul A. Smith asked the following question. If a finite group $G$ acts smoothly on a sphere with exactly two fixed points, is it true that the tangent $G$–modules at the two points are always isomorphic? We focus on the case $G$ is an Oliver group and we present a classification of finite Oliver groups $G$ with Laitinen number $a_G=0$ or $1$. Then we show that the Smith Isomorphism Question has a negative answer and $a_G\ge 2$ for any finite Oliver group $G$ of odd order, and for any finite Oliver group $G$ with a cyclic quotient of order $pq$ for two distinct odd primes $p$ and $q$. We also show that with just one unknown case, this question has a negative answer for any finite nonsolvable gap group $G$ with $a_G\ge 2$. Moreover, we deduce that for a finite nonabelian simple group $G$, the answer to the Smith Isomorphism Question is affirmative if and only if $a_G=0$ or $1$.