From: "J. Mayer" <jm@hadiko.de>
Newsgroups: sci.math
Subject: Re: Q: What is a free group?
Date: Fri, 10 Jul 1998 23:25:56 +0200

Norm Dresner schrieb:

> A book I was reading defined a
>                 Free Abelian Group of Rank R
> and later said
>                 Suppose G is not a free group
>
> I know what Abelian means, but what exactly makes a group "free"?
>
>         Norm D.

 First of all, you have to distinguish between a free group and a free
abelian group.... And then there are several ways of looking at things:

1. A free group is a group without relations- that means that it is
generated by a certain number of elements (usually finite) and every
equation that you could possibly write using group elements (that is
product of the generators or their inverses) or for that matter any
equation that you could write using only the generators themselves with
their inverses (such as abc=1 or a^n=1 or ababababa=cdedde, etc.) is
FALSE. The notable exception here is, that "trivial equations" such as
a*a^-1=1 which must be true in any group must of course be true here as
well. But all other equations must be false. (BTW, here the group is
written multiplicatively with 1 being the neutral element). Another way
of saying it: A group being free means that there is no non-trivial
expression that equal 1 (notice the similarity to free i.e. linearly
independent in the theory of vector spaces!).

If the free group (of rank n) is finitely generated by n Elements (say
a,b,c,d,...) then the group consists of all expressions of these
elements and their inverses and all distict expressions are unequal
(because otherwise you would have an equation). Again, expressions which
are trivially equal such as a*b*b^-1*a*c=a^2*c aren't considered!
Particularly, the group with one generator is isomorphic to Z. In a
sense, a free group is the set of all words that you can write using an
alphabet consisting of the generators and their inverses. Examples of
groups that aren't free are all finite groups because you have the
equation g^n=1 where n is the order of the group. Similarly, a group
that has a finite subgroup can't be free because it has the same
equation. As a matter of fact, there are only a few free groups because
they are isomorphic iff they have the same (minimal) number of
generators- this number is called the rank.

2. A free abelian group is the same thing, except that the group is
abelian- i.e. you have allow additionally to the trivial equations above
all equations of the form ab=ba. The free abelian group of rank n is
isomorphic to Z^n.

Hope that helps.