## Journal of Mathematics of Kyoto University

- J. Math. Kyoto Univ.
- Volume 48, Number 1 (2008), 219-227.

### The Smith sets of finite groups with normal Sylow 2-subgroups and small nilquotients

Akihiro Koto, Masaharu Morimoto, and Yan Qi

#### Abstract

The Smith equivalence of real representations of a finite group has been studied by many mathematicians, e.g. J. Milnor, T. Petrie, S. Cappell-J. Shaneson, K. Pawałowski-R. Solomon. For a given finite group, let the primary Smith set of the group be the subset of real representation ring consisting of all differences of pairs of prime matched, Smith equivalent representations. The primary Smith set was rarely determined for a nonperfect group G besides the case where the primary Smith set is trivial. In this paper we determine the primary Smith set of an arbitrary Oliver group such that a Sylow 2-subgroup is normal and the nilquotient is isomorphic to the direct product of a finite number of cyclic groups of order 2 or 3. In particular, we answer to a problem posed by T. Sumi.

#### Article information

**Source**

J. Math. Kyoto Univ., Volume 48, Number 1 (2008), 219-227.

**Dates**

First available in Project Euclid: 14 August 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.kjm/1250280981

**Digital Object Identifier**

doi:10.1215/kjm/1250280981

**Mathematical Reviews number (MathSciNet)**

MR2437897

**Zentralblatt MATH identifier**

1157.55004

**Subjects**

Primary: 55M35: Finite groups of transformations (including Smith theory) [See also 57S17]

Secondary: 57S17: Finite transformation groups 57S25: Groups acting on specific manifolds 20C15: Ordinary representations and characters

#### Citation

Koto, Akihiro; Morimoto, Masaharu; Qi, Yan. The Smith sets of finite groups with normal Sylow 2-subgroups and small nilquotients. J. Math. Kyoto Univ. 48 (2008), no. 1, 219--227. doi:10.1215/kjm/1250280981. https://projecteuclid.org/euclid.kjm/1250280981