  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 > 1 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
> 1. 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.
