## Taiwanese Journal of Mathematics

### Abelian $p$-Groups of Symmetries of Surfaces

Y. Talu

#### Abstract

An integer $g \geq 2$ is said to be a genus of a finite group $G$ if there is a compact Riemann surface of genus $g$ on which $G$ acts as a group of automorphisms. In this paper finite abelian $p$-groups of arbitrarily large rank, where $p$ is an odd prime, are investigated. For certain classes of abelian $p$-groups the minimum reduced stable genus $\sigma_0$ of $G$ is calculated and consequently the genus spectrum of $G$ is completely determined for certain "extremal" abelian $p$-groups. Moreover for the case of $Z_p^{r_1} \oplus Z_{p^2}^{r_2}$ we will see that the genus spectrum determines the isomorphism class of the group uniquely.

#### Article information

Source
Taiwanese J. Math., Volume 15, Number 3 (2011), 1129-1140.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406290

Digital Object Identifier
doi:10.11650/twjm/1500406290

Mathematical Reviews number (MathSciNet)
MR2829902

Zentralblatt MATH identifier
1234.57022

#### Citation

Talu, Y. Abelian $p$-Groups of Symmetries of Surfaces. Taiwanese J. Math. 15 (2011), no. 3, 1129--1140. doi:10.11650/twjm/1500406290. https://projecteuclid.org/euclid.twjm/1500406290

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