From: mareg@mimosa.csv.warwick.ac.uk ()
Subject: Re: Groups
Date: 13 Oct 2000 10:05:41 GMT
Newsgroups: sci.math
Summary: [missing]

In article <39E62B6C.D9B2A9C6@club-internet.fr>,
	n_sotto@club-internet.fr writes:
>Hi,
>let G be a group generated by a and b. If a, b and (ab) have finite
>periods, is G a finite group?
>Thanks, Nicola.
>

Not necessarily. Let the orders (periods) of a, b (ab) be l, m, n. Then
G is necessarily finite if and only if  1/l + 1/m + 1/n  > 1 - i.e.
(assuming, l,m,n > 1) iff {l,m,n} = {2,2,n}, {2,3,4} or {2,3,5}.

In the cases {2,3,6}, {2,4,4} amd {3,3,3} where 1/l + 1/m + 1/n = 1,
you get infinite symmetry groups of regular tesselations of the Euclidean
planes as examples.

In general, it is possible for *all* elements of a group generated by a,b, to
have finite orders but for the group to be infinite.

Derek Holt.