Step-by-step explanation:
Say [tex]a[/tex] is an element of [tex]G[/tex] which might have more than 1 inverse. Let's call them [tex]b[/tex], and [tex]c[/tex]. So that [tex]a[/tex] has apparently two inverses, [tex]b[/tex] and [tex]c[/tex].
This means that [tex]a*b = e[/tex] and that [tex]a*c=e[/tex](where [tex]e[/tex] 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
[tex]a*b=a*c[/tex]
And operating both sides by b by the left, we'd get:
[tex]b*(a*b)=b*(a*c)[/tex]
Now, remember the operation on any group is associative, meaning we can rearrange the parenthesis to our liking, gettting then:
[tex](b*a)*b=(b*a)*c[/tex]
And since b is the inverse of a, [tex]b*a=e[/tex], and so:
[tex](e)*b=(e)*c[/tex]
[tex]b=c[/tex] (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.
