Editing Group (mathematics) (section)

=== Uniqueness of inverses ===
The group axioms also imply that the inverse of each element is unique. Let a group element <math>a</math> have both <math>b</math> and <math>c</math> as inverses. Then
: <math>\begin{align}
b &=  b\cdot e 
    && \text{(}e \text { is the identity element)}\\
  &=  b\cdot (a \cdot c) 
    && \text{(}c \text { and } a \text{ are inverses of each other)}\\
  &=  (b\cdot a) \cdot c 
    && \text{(associativity)}\\
  &=  e \cdot c 
    && \text{(}b \text { is an inverse of } a\text{)}\\
  &=  c 
    && \text{(}e \text { is the identity element and } b=c\text{)}
\end{align}</math>

Therefore, it is customary to speak of ''the'' inverse of an element.{{sfn|Lang|2005|loc=§II.1|p=17}}