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
![a*b=a*c](https://tex.z-dn.net/?f=a%2Ab%3Da%2Ac)
And operating both sides by b by the left, we'd get:
![b*(a*b)=b*(a*c)](https://tex.z-dn.net/?f=b%2A%28a%2Ab%29%3Db%2A%28a%2Ac%29)
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](https://tex.z-dn.net/?f=%28b%2Aa%29%2Ab%3D%28b%2Aa%29%2Ac)
And since b is the inverse of a,
, and so:
![(e)*b=(e)*c](https://tex.z-dn.net/?f=%28e%29%2Ab%3D%28e%29%2Ac)
(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.