__Example 1__**:** If be a normal subgroup of a finite group , then prove that

__Solution__**:** number of distinct right (or left) cosets of in , as is the collection of all right (or left) cosets of in

by Lagrange’s Theorem

__Example 2__**:** Show that every quotient group of a cyclic group is cyclic but not conversely.

__Solution__**:**

Let be a subgroup of a cyclic group . Then is also cyclic because every cyclic group is abelian. Therefore is a normal subgroup is .

Let be a generator of and be any element of , where is some integer. Then is any element of .

Also, it can be proved easily that , for every integer . Therefore, is cyclic and its generator is .

Its converse is not true, for example if and be the symmetric and alternating groups on the three symbols then the quotient group is cyclic, whereas is not.

__Example 3__**:** Show that every quotient group of an abelian group is abelian but its converse is not true.

__Solution__**:**

Let be arbitrary, then are any two elements of the quotient group . Then we have

Therefore, is an abelian.

Its converse is not true, for example if and be the symmetric and alternating groups on the three symbols then the quotient group being of order

**2**is abelian whereas is not.