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

Step-by-step 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.