Question about the answer to Kac's problem: 'Can one hear the shape of a drum?'.

Question

I'm looking at the article of Gordon, Webb and Wolpert http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, having only basic notions of group theory. In this article the authors describe in a simple way the construction of a pair of isospectral but not isometric plane domains.

At pag $52$ (paragraph entitled 'One can't hear the shape of a drum') it's written that they use the group $G$ used by Robert Brooks and Peter Buser to construct isospectral surfaces. I thought that this $G$ was $PSL(3,2)$ but then it is said that $G$ is generated by three elements $\alpha, \beta$ and $\gamma$, while $PSL(3,2)$ is generated by two elements.

Can someone tell me who are explicitly this $G$ and the generators $\alpha$, $\beta $ and $\gamma$? Thanks!

Answer 1

in GAP we get:

G:=Group((1,2)(5,6),(2,4)(3,5),(2,6)(3,7));;
StructureDescription(G);
"PSL(3,2)"

A group can be generated by a different number of generators. The symmetric group $S_n$ is generated by a cyclic permutation of order $n$ and a permutation of two elements (e.g. $(1..n)$ and $(1,2)$). In this case there are only two generators, but the group can also be generated by all the permutations of two elements giving $n(n+1)/2$ generators.