Abelian Groups

Date: 10/22/2003 at 09:07:50
From: Blake
Subject: abstract algebra

Let G be a group with the identity element e. Show that:

1) if x^2 = e for all x in G, then G is Abelian;
2) if (xy)^2 = x^2 * y^2 for all x,y in G, then G is Abelian.

I know (or think) that in both parts of the problem, x*x = e = x*x 
would demonstrate commutativity (the fourth requirement for groups 
which make them Abelian), and I'm thinking that it would have to be 
something to the effect of x^2 * y^2 = y^2 * x^2 for the second part
to indicate commutativity for that part...

Date: 10/22/2003 at 10:14:21
From: Doctor Luis
Subject: Re: abstract algebra

Hi Blake,

To prove commutativity, you need to show that xy = yx for any x,y in G.

For part (1), take any x,y in G; then xy is also in G.  This means
three things: x^2 = e, y^2 = e, (xy)^2 = e.  Use this last equation
to prove commutativity (xy = yx).  Recall that (xy)^2 = xyxy.

For part (2), we start the same way: take any x,y in G.  Then by
hypothesis, we know that (xy)^2 = x^2 * y^2.  Then, we have xyxy =
xxyy.  Now, since G is a group, there exist inverses x',y' in G such
that x'x = e, and yy' = e.  Use these inverses to show that xyxy =
xxyy implies commutativity.

I hope this helped! Let us know if you have any more questions.

- Doctor Luis, The Math Forum
  http://mathforum.org/dr.math/