Properties of a Group

The identity element of a 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. .