## Smith equivalence of representations for finite perfect groups

### Authors:

- Erkki Laitinen,
- Krzysztof Pawałowski

### 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 > 1 and primes p, q, we prove similar results for the group G = A_{n}, SL_{2}(F_{P}), or PSL_{2}(F_{q}). In particular, G has Smith equivalent representations that are not isomorphic if and only if n ≥ 8, p ≥ 5, q ≥ 19. ©1999 American Mathematical Society.

- Record ID
- UAM4c8590b85786400eafd16f4402bddf58
- Author
- Journal series
- Proceedings of the American Mathematical Society, ISSN 0002-9939
- Issue year
- 1999
- Vol
- 127
- Pages
- 297-307
- ASJC Classification
- ;
- Language
- en English
- Score (nominal)
- 25
- Score source
- journalList
- Publication indicators
- = 12.000; : 2014 = 1.093; : 2006 = 0.513 (2) - 2007=0.611 (5)

- Uniform Resource Identifier
- https://researchportal.amu.edu.pl/info/article/UAM4c8590b85786400eafd16f4402bddf58/

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