# Homework Help: Introduction to Group Theory - Abstract Algebra

1. Jan 16, 2009

### descendency

1. The problem statement, all variables and given/known data
Prove that if (ab)2 = a2b2 in a group G, then ab = ba.

2. Relevant equations
* For each element a in G, there is an element b in G (called the inverse of a) such that ab = ba = e (the identity).
* For each element in G, there is a unique element b in G such that ab = ba = e.
* The operation (under which the group is defined) is associative; that is, (ab)c = a(bc).

3. The attempt at a solution
$$(ab)^{2} = (a)^{2}(b)^{2}$$
$$(a)^{-1}(ab)^{2}(b)^{-1} = (a)^{-1}(a)^{2}(b)^{2}(b)^{-1}$$
$$(a)^{-1}abab(b)^{-1} = (a)^{-1}aabb(b)^{-1}$$
ebae = eabe
ba = ab.

Is this allowed? (Keep in mind, the operation which the group is defined over is not necessarily multiplication. it may be composition.)

Last edited: Jan 16, 2009
2. Jan 16, 2009

### NoMoreExams

Sure is but your pre last line should be ba = ab I believe.

Also your 2nd "given" is the same as your first, you probably meant something else?

3. Jan 16, 2009

### descendency

The second given is the uniqueness statement. The first given is the rule that an element in a group has at least an inverse (not necessarily unique). I said "the" when I meant "an".

edit: Thanks for correcting my mistake. It's a typo. I had it right on paper, I just didn't type out what I wrote down for some reason. . .

Last edited: Jan 16, 2009
4. Jan 16, 2009

### NoMoreExams

Then that seems redundant, line 2 is stronger than line 1.

5. Jan 16, 2009

### Unco

Your work is fine, where I read it as (line 1) => (line 2) => ... .

However, depending on the level of pendency of your instructor you may be required to justify your use of associativity. E.g., the left-hand side is $$(ab)^2 = (ab)(ab) = a(b(ab)) = a((ba)b)$$, and the right-hand side is $$a^2b^2 = (aa)(bb) = a(a(bb)) = a((ab)b)$$; then multiply both sides on the left by $$a^{-1}$$, and so on.