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Abh. Math. Sem. Univ. Hamburg
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
\Gammam(q) be
the group generated by two elements x and y subject to the
relations x2m=yqm=1 and
x2=yq; that is,
\Gammam(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
\Gammam(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 \Gammam(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\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 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).
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