Uniqueness of inverses - Alex's public notes

Lemma

Suppose we have some set $X$ and associative binary operation $\cdot$ with a unit $1$ such that $x \cdot 1 = 1 \cdot x = x$ then two-sided inverses if they exists are unique.

Proof

Suppose for some element $x \in X$ and let $x_1^{-1}$ and $x_2^{-1}$ be two-sided inverses. Then observe

$$ x_1^{-1} = x_1^{-1} \cdot 1 = x_1^{-1} \cdot ( x \cdot x_2^{-1}) = (x_1^{-1} \cdot x) \cdot x_2^{-1} = 1 \cdot x_2^{-1} = x_2^{-1} $$

giving equality.