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2014•Glasgow Mathematical JournalOpen access

COMPUTING WITH SUBGROUPS OF THE MODULAR GROUP

Markus Kirschmer, C. R. Leedham-Green

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Abstract

Abstract We give several algorithms for finitely generated subgroups of the modular group PSL2(ℤ) given by sets of generators. First, we present an algorithm to check whether a finitely generated subgroup H has finite index in the full modular group. Then we discuss how to parametrise the right cosets of H in PSL2(ℤ), whether the index is finite or not. Further, we explain how an element in H can be written as a word in a given set of generators of H.

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Abstract We give several algorithms for finitely generated subgroups of the modular group PSL2(ℤ) given by sets of generators. First, we present an algorithm to check whether a finitely generated subgroup H has finite index in the full modular group. Then we discuss how to parametrise the right cosets of H in PSL2(ℤ), whether the index is finite or not. Further, we explain how an element in H can be written as a word in a given set of generators of H.

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Available abstract

Abstract We give several algorithms for finitely generated subgroups of the modular group PSL2(ℤ) given by sets of generators. First, we present an algorithm to check whether a finitely generated subgroup H has finite index in the full modular group. Then we discuss how to parametrise the right cosets of H in PSL2(ℤ), whether the index is finite or not. Further, we explain how an element in H can be written as a word in a given set of generators of H.

Key concepts: Coset, PSL, Mathematics, Modular group, Modular design, Generating set of a group, Group (periodic table), Word (group theory)

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