Smith equivalence of representations for finite perfect groups
Proc. Amer. Math. Soc. 127 (1999), 297-307.

Erkki Laitinen 
Faculty of Mathematics and Computer Science, 
Adam Mickiewicz University of, 
Poznan, ul. Jana Matejki 48/49, PL-60-769 Poznan, Poland 
and
Krzysztof Pawalowski
Faculty of Mathematics and Computer Science, Adam Mickiewicz 
University of, Poznan, ul. Jana Matejki 48/49, PL-60-769 Poznan, Poland 


Abstract. Using smooth one-fixed-point actions on spheres and a result due to 
Bob Oliver on the tangent representations at fixed points for smooth group 
actions on disks, we obtain a similar result for perfect group actions on 
spheres. For a finite group $G$, we compute a certain subgroup $IO'(G)$ of the 
representation ring $RO(G)$. This allows us to prove that a finite perfect 
group $G$ has a smooth $2$-proper action on a sphere with isolated fixed 
points at which the tangent representations of $G$ are mutually nonisomorphic 
if and only if $G$ contains two or more real conjugacy classes of elements not 
of prime power order. Moreover, by reducing group theoretical computations to 
number theory, for an integer $n \ge 1$ and primes $p$, $q$, we prove similar 
results for the group $G=A_n$, ${\rm SL}_2(\mathbb F_p)$, or 
${\rm PSL}_2(\mathbb F_q)$. In particular, $G$ has Smith equivalent 
representations that are not isomorphic if and only if $n \ge 8$, $p \ge 5$,
$q \ge 19$.