(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that if (ab)^{2}= a^{2}b^{2}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

[tex](ab)^{2} = (a)^{2}(b)^{2}[/tex]

[tex](a)^{-1}(ab)^{2}(b)^{-1} = (a)^{-1}(a)^{2}(b)^{2}(b)^{-1}[/tex]

[tex](a)^{-1}abab(b)^{-1} = (a)^{-1}aabb(b)^{-1}[/tex]

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.)

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# Homework Help: Introduction to Group Theory - Abstract Algebra

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