Congruence

From Farey symbols to generators for subgroups of finite index in integral group rings of finite groups. E. Jespers, G. Leal [in Manuscr. Math. 78, No. 3, 303-315 (1993; Zbl 0802.16025)] proved that Bass cyclic units and bicyclic units generate a subgroup of finite index in the integral group ring of a finite group G provided G does not have a fixed-point free nonabelian epimorphic image and there are no exceptional Wedderburn components, that is, 2×2 matrix rings over either the rationals, a quadratic imaginary extension of the rationals or a non-commutative division algebra, in its rational group algebra. The aim of the paper is to remove the exception of 2×2 matrix rings over the rationals introducing finitely many of the so called Farey units as generators (themselves generating free subgroups) to obtain a subgroup of finite index. Farey units are constructed by means of Farey symbols, which are in one-to-one correspondence with fundamental polygons of congruence subgroups of PSL 2 (ℤ). The classical theory, due to Poincaré, for producing generators and relations for subgroups of the modular group PSL 2 (ℤ), acting on the hyperbolic plane, by means of a fundamental polygon, together with introduction of Farey symbols representing subgroups of finite index in PSL 2 (ℤ), due to R. S. Kulkarni [Am. J. Math. 113, No. 6, 1053-1133 (1991; Zbl 0758.11024)], are described in detail in the paper. The algorithm devised here is implemented in the package Congruence for the computer algebra system GAP by the authors.