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Some results on quotients of triangle groups. (English) Zbl 0546.20027
The author studies finite permutation representations of triangle groups. He presents graphical and algorithmic computational techniques for handling them. The basic tools are coset diagrams generalizing the Cayley graph associated with a finite presentation for group $$G$$. The algorithms derive from well known procedures from combinatorial group theory. The Todd-Coxeter algorithm and the Schreier-Sims algorithm provide a basis for explicit machine computation by which those finite quotients of a triangle group $$G$$ are found which allow transitive permutation representations of small degree.
New results on minimal generating pairs of the groups $$A_ n$$, $$S_ n$$, and of the group of Rubik’s cube are given. We cite as an example: Theorem: Given $$m>6$$, all but finitely many alternating groups $$A_ n$$ occur as quotients of $$\Delta(2,3,m)$$. Moreover, if $$m$$ is even, then all but finitely many symmetric groups $$S_ n$$ occur as well. Using results of Tucker, the strong symmetric genus of all $$A_ n$$ and $$S_ n$$ is determined. For some groups upper bounds for the genus of the group are given via minimal generating pairs and embedding some Cayley graph of $$G$$ into a surface whose genus is known from the Riemann-Hurwitz equation.
Reviewer: H.R.Schneebeli

MSC:
 20F05 Generators, relations, and presentations of groups 20B35 Subgroups of symmetric groups 20-04 Software, source code, etc. for problems pertaining to group theory 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures 20C30 Representations of finite symmetric groups
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