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.