Parity Patterns Associated with Lifts of Hecke Groups

Let q be an odd prime, m a positive integer, and let Γm(q) be the group generated by two elements x and y subject to the relations x = y = 1 and x = y; that is, Γm(q) is the free product of two cyclic groups of orders 2m respectively qm, amalgamated along their subgroups of order m. Our main result determines the parity behaviour of the generalized subgroup numbers of Γm(q) which were defined in [T. W. Müller, Adv. in Math. 153 (2000), 118–154], and which count all the homomorphisms of index n subgroups of Γm(q) into a given finite group H , in the case when gcd(m, |H |) = 1. This computation depends upon the solution of three counting problems in the Hecke group H(q) = C2 ∗Cq: (i) determination of the parity of the subgroup numbers of H(q); (ii) determination of the parity of the number of index n subgroups of H(q) which are isomorphic to a free product of copies of C2 and of C∞; (iii) determination of the parity of the number of index n subgroups in H(q) which are isomorphic to a free product of copies of Cq. The first problem has already been solved in [T. W. Müller, in: Groups: Topological, Combinatorial and Arithmetic Aspects, (T. W. Müller ed.), LMS Lecture Notes Series 311, Cambridge University Press, Cambridge, 2004, pp. 327–374]. The bulk of our paper deals with the solution of Problems (ii) and (iii).