Division Laws for Groups

Theorem
Let $$G$$ be a group.

Let $$a, b, x \in G$$.

Then:
 * $$a x = b \iff x = a^{-1} b$$
 * $$x a = b \iff x = b a^{-1}$$

Proof
All derivations can be achieved using applications of the group axioms.

Proof of First Result
$$ $$ $$ $$

and the converse:

$$ $$ $$ $$

Proof of Second Result
$$ $$ $$ $$

and the converse:

$$ $$ $$ $$