What is wrong with my proof here?
Proof:
Let a,b be elements of a group and let aa=b.
Through manipulation, we see that
a=ba−1
b−1a=a−1
b−1aa−1=e
b−1=e
We of course know that
bb−1=e
And from above, we see that
be=e
or
b=e.
And since aa=b, we can see that aa=e and that a=a−1.
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