
The identity element of a group is unique.

The inverse of each element of a group is unique, i.e. in a group $$G$$ with operation $$ * $$ for every $$a \in G$$, there is only element $${a^{ – 1}}$$ such that$${a^{ – 1}} * a = a * {a^{ – 1}} = e$$, $$e$$ being the identity.

The inverse $$a$$ of$${a^{ – 1}}$$, then the inverse of $${a^{ – 1}}$$ is $$a$$, i.e. $${\left( {{a^{ – 1}}} \right)^{ – 1}} = a$$.

The inverse of the product of two elements of a group $$G$$ is the product of the inverse taken in the inverse order, i.e. $${\left( {a * b} \right)^{ – 1}} = {b^{ – 1}} * {a^{ – 1}}{\text{ }}\forall a,b \in G$$.

Cancellation laws holds in a group, i.e. if $$a,b,c$$ are any elements of a group $$G$$, then $$a * b = a * c \Rightarrow b = c$$ (left cancellation law), $$b * a = c * a \Rightarrow b = c$$ (right cancellation law).

If $$G$$ is a group with binary operation $$ * $$ and if $$a$$ and $$b$$ are any elements of $$G$$, then the linear equations $$a * x = b$$ and $$y * a = b$$ have unique solutions in $$G$$.

The left inverse of an element is also its right inverse, i.e. $${a^{ – 1}} * a = e = a * {a^{ – 1}}$$.