Prove that the each element of a group g has a unique inverse. that is, if a, b, w e g satisfy then b

Step-by-step explanation:

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

This means that and that (where 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

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

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

And since b is the inverse of a, , and so:

(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.