From: mareg@mimosa.csv.warwick.ac.uk ()
Subject: Re: Free group with 2 generators
Date: 3 Jan 2001 11:57:33 GMT
Newsgroups: sci.math
Summary: Number of subgroups of index n in the free group of rank r

In article <92t00a$3n5$1@nnrp1.deja.com>,
	tim_brooks@my-deja.com writes:
>Let G be the free group on 2 generators a and b.
>If d >=1 how many subgroups of index d there are in G ?

Theorem 7.2.9 of Marshall Hall, "The Theory of Groups":

The number N_{n,r} of subgroups of index n in the free group of rank r is
given recursively by

N_{1,r} = 1

N_{n,r} = n(n!)^{r-1} - \sum_{i=1}^{n-1} (n - i)!^{r-1} N_{i,r}.o

In particular, the values of N_{n,2} for n=1,2,3,... are

1, 3, 13, 71, 461, 3447, 29093, 273343, 2829325, 31998903, 392743957, ...

Derek Holt.