# Properties of a Group

1. The identity element of a group is unique.
2. 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.
3. The inverse $a$ of${a^{ – 1}}$, then the inverse of ${a^{ – 1}}$ is $a$, i.e. ${\left( {{a^{ – 1}}} \right)^{ – 1}} = a$.
4. 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. ${\left( {a * b} \right)^{ – 1}} = {b^{ – 1}} * {a^{ – 1}}{\text{ }}\forall a,b \in G$.
5. 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).
6. 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 $G$.
7. The left inverse of an element is also its right inverse, i.e. ${a^{ – 1}} * a = e = a * {a^{ – 1}}$.