
The identity element of s group is unique.

The inverse of each element of a group is unique, i.e., in a group with operation for every , there is only element such that, being the identity.

The inverse of, then the inverse of is , i.e., .

The inverse of the product of two elements of a group is the product of the inverse taken in the inverse order i.e..

Cancellation laws holds in a group, i.e., if are any elements of a group , then (left cancellation law), (right cancellation law).

If is a group with binary operation and if and are any elements of , then the linear equations and have unique solutions in .

The left inverse of an element is also its right inverse, i.e. .