Examples Subgroup of Cyclic Groups

Example 1: Find the proper subgroups of the multiplicative group G of the sixth roots of unity.
Solution: From trigonometry we know that six sixth roots of unity are

\cos  \left( {\frac{{n\pi }}{3}} \right) + i\sin \left( {\frac{{n\pi }}{3}} \right)


{e^{\frac{{n\pi i}}{3}}}

where n = 0,1,2,3,4,5

\therefore  \,\,\,G = \left\{ {{e^{\frac{{n\pi i}}{3}}},\,n = 0,1,2,3,4,5} \right\}

If {H_1} and {H_2} be its proper subgroups, then

{H_1}  = \left\{ {{e^{\frac{{n\pi i}}{3}}},\,n = 3,6} \right\} = \left\{ {{e^{\pi  i}},\,{e^{2\pi i}} = 1} \right\} = \left\{ {{e^{\pi i}},\,1} \right\}


{H_2}  = \left\{ {{e^{\frac{{n\pi i}}{3}}},\,n = 2,4,6} \right\} = \left\{  {{e^{\frac{{2\pi i}}{3}}},\,{e^{\frac{{4\pi i}}{3}}},1} \right\}

Example 2: Find all the subgroup of a cyclic group of order 12.
Solution: We know that the integral divisors of 12 are 1, 2, 3, 4, 6, 12. Now, there exists one and only one subgroup of each of these orders. Let a be the generators of the group and m be a divisor of 12. Then there exists one and only one element in G whose order is m, i.e. {a^{\frac{{12}}{m}}}.
All the elements of order 1, 2, 3, 4, 6, 12 will give subgroups.
\therefore \,\left( {{a^{12}}} \right) = \left\{ e  \right\},\left( {{a^6}} \right),\left( {{a^4}} \right),\left( {{a^3}}  \right),\left( {{a^2}} \right),\left( a \right) are the required subgroups.

Example 3:
(i) Can an abelian groups have a non-abelian subgroup?
(ii) Can a non-abelian group have an abelian subgroup?
(iii) Can a non-abelian group have a non-abelian subgroup?
(i) Every subgroup of an abelian group is abelian. If G is an abelian group and H is a subgroup of G, then the operation on H is commutative because it is already commutative in G and H is a subset of G. Hence an abelian group cannot have a non-abelian subgroup.
(ii) A non-ableian group can have an abelian subgroup. For example, the symmetric group {P_3} of permutation of degree 3 is non-abelian while its subgroup {A_3} is ableian.
(iii) A non-abelian group can have a non-abelian subgroup. For example, {P_4} is a non-abelian group and its subgroup {A_4} is also non-abelian.