Parity patterns associated with lifts of Hecke groups

Details Parity patterns associated with lifts of Hecke groups Parity patterns associated with lifts of Hecke groups

2008-03-28

arxiv.org/abs/0803.4175v2

Abh. Math. Sem. Univ. Hamburg 78 (2008), 99-147.

Mathematics

Group Theory

AmS-LaTeX; 49 pages; minor corrections, Section 5 restructured for better reading (with some proofs put in a new appendix), ne

Scientific paper

Let $q$ be an odd prime, $m$ a positive integer, and let $\Ga_m(q)$ 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, $\Ga_m(q)$ 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 $\Ga_m(q)$ which were defined in [T. W. M\"uller, Adv. in Math. 153 (2000), 118-154], and which count all the homomorphisms of index $n$ subgroups of $\Ga_m(q)$ 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 $\mathfrak H(q)=C_2*C_q$: (i) determination of the parity of the subgroup numbers of $\mathfrak H(q)$; (ii) determination of the parity of the number of index $n$ subgroups of $\mathfrak H(q)$ 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 $\mathfrak H(q)$ which are isomorphic to a free product of copies of $C_q$. The first problem has already been solved in [T. W. M\"uller, in: {\it Groups: Topological, Combinatorial and Arithmetic Aspects}, (T. W. M\"uller 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).

Affiliated with

Also associated with

No associations

Rating

Parity patterns associated with lifts of Hecke groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Parity patterns associated with lifts of Hecke groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Parity patterns associated with lifts of Hecke groups will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-174386