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Homework Statement
Prove that if (ab)2 = a2b2 in a group G, then ab = ba.
Homework 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).
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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