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
.
Keywords
finite group, Oliver group, Laitinen number, smooth action,
sphere, tangent module, Smith equivalence, Laitinen–Smith
equivalence