# Examples of Quotient Groups

__Example 1__**:** If is 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 of .

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

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

Its converse is not true; for example if and are the symmetric and alternating groups of 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 are the symmetric and alternating groups of the three symbols then the quotient group being of order **2** is abelian whereas is not.