Verify the multiplication of two absolute abelian group elements

Question
Given that aa and bb belong to an Abelian group and that |a|=6|a|=6 and |b|=6|b|=6, what can you say about |ab||ab|?

My solution
|a|=6a=6a6=e
|a|=6 \\
a=6 \\
a^6=e

similarly:
|b|=6b=6b6=e
|b|=6 \\
b=6 \\
b^6=e

hence
|ab|=(ab)(ab)=(ab)6=a6b6=ee=e
|ab|=(ab) \\
(ab)=(ab)^6=a^6b^6=ee=e

therefore, |ab|=6|ab|=6, and abab belongs to GG.

Q: Pls what sis I do wrong in answering the question???

Thanks

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2 Answers
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Saying that |a|=6|a|=6 means that a6=ea^6=e and ak≠ea^k\ne e, for 0