Question

Prove that the each element of a group G has a unique inverse. That is, if a, b, W E G satisfy then b

Answer 1

Explanation:

Say
`a` is an element of
`G` which might have more than 1 inverse. Let's call them
`b`, and
`c`. So that
`a` has apparently two inverses,
`b` and
`c`.

This means that
`a*b = e` and that
`a*c=e`(where
`e` is the identity element of the group, and * is the operation of the group)

But so we could merge those two equations into a single one, getting

`a*b=a*c`

And operating both sides by b by the left, we'd get:

`b*(a*b)=b*(a*c)`

Now, remember the operation on any group is associative, meaning we can rearrange the parenthesis to our liking, gettting then:

`(b*a)*b=(b*a)*c`

And since b is the inverse of a,
`b*a=e`, and so:

`(e)*b=(e)*c`

`b=c` (since e is the identity of the group)

So turns out that b and c, which we thought might be two different inverses of a, HAVE to be the same element. Therefore every element of a group has a unique inverse.