Inverse of a group element is unique

Dependencies:

  1. Group Used in definition

Inverse of a group element is unique.

Proof

Let $l_i$ be the $i^{\textrm{th}}$ left inverse and $r_i$ be the $i^{\textrm{th}}$ right inverse.

\begin{align} l_i &= l_i * e \tag{$e$ is identity} \\ &= l_i * (a * r_j) \tag{$r_j$ is right inverse} \\ &= (l_i * a) * r_j \tag{associativity} \\ &= e * r_j \tag{$l_i$ is left inverse} \\ &= r_j \tag{$e$ is identity} \end{align}

This means that every left inverse is equal to every right inverse. This means that they're all equal.

Dependency for:

  1. Conditions for a subset to be a subgroup
  2. gH = H iff g in H

Info:

Transitive dependencies:

  1. Group