Properties of Group

• The identity element of a group is always unique.
• The inverse of each element of a group is unique, i.e., in a group G with operation * for every, there is only element such that, e being the identity.
• The inverse a of, then the inverse of  is a , i.e., .
• 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..
• Cancellation laws holds in a group, i.e., if  are any elements of a group G, then  (left cancellation law),  (right cancellation law).
• If G  is a group with binary operation and if  and  are any elements of G, then the linear equations  and  have unique solutions in G.
• The left inverse of an element is also its right inverse, i.e. .