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.