Abelian GroupsDate: 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/ |
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