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**78** (2008), 99-147,
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# Parity patterns associated with lifts of Hecke groups

### (45 pages)

**Abstract.**
Let *q* be an odd prime, *m* a positive integer, and let
*\Gamma*_{m}(*q*) be
the group generated by two elements *x* and *y* subject to the
relations *x*^{2m}=*y*^{qm}=1 and
*x*^{2}=*y*^{q}; that is,
*\Gamma*_{m}(*q*) is the
free product of two cyclic groups of orders 2*m* respectively *qm*,
amalgamated along their subgroups of order *m*. Our main result
determines the parity behaviour of the generalized subgroup numbers of
*\Gamma*_{m}(*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 *\Gamma*_{m}(*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*)=*C*_{2}**C*_{q}:
(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 *C*_{2} and of *C*_{\infty};
(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 *C*_{q}. 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).

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