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.
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.
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.