> < ^ Date: Tue, 04 Oct 1994 12:34:00 BST
> ^ From: Jim Howie <j.howie@ma.hw.ac.uk >
> < ^ Subject: Re: GAP computation

In connection with a GAP computation involving the Mathieu
groups M24 and M23, Derek Holt asked for theoretical results
giving the rank of a free subgroup of a free product.
Such a result exists for the finite index case, when the
free factors are finite (which is the case in his example).

Use rational euler characteristics.   For a finite group
G, this is defined as \chi(G)=1/|G|.   For a free product
you have \chi(A*B)=\chi(A)+\chi(B)-1.   Hence if G is the
free product of three cyclic groups of order 3 we have
\chi(G)=1/3 + 1/3 + 1/3 - 2 = -1.   For subgroups of finite
index H\subseteq G we have the rule \chi(H)=|G:H|.\chi(G),
so for the kernel K of Derek's map G --> M23 we have
\chi(K)=|M23|.\chi(G) = -|M23|.
Finally, the euler characteristic of a free group of rank r
is \chi(F_r)=1-r, so we recover Derek's result:
rank(K)=1-\chi(K)=|M23|+1.


QED.

Jim Howie.

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