Proving Cyclic Nature of Subgroups in Groups of Order pq

When dealing with a group ( G ) of order ( pq ), where ( p ) and ( q ) are distinct prime numbers, it is crucial to prove that any nontrivial subgroup of ( G ) is cyclic. In this article, we will present a step-by-step approach using Sylow's Theorems and properties of groups of small order. This method ensures a rigorous and comprehensive understanding of the problem.

Step 1: Identifying Sylow Subgroups

The first step is to identify the Sylow subgroups of ( G ). For a group ( G ) with order ( pq ), we denote the Sylow ( p )-subgroups and Sylow ( q )-subgroups as follows:

Let ( n_p ) be the number of Sylow ( p )-subgroups of ( G ) and ( n_q ) the number of Sylow ( q )> By Sylow's Theorems: ( n_p equiv 1 mod p ) and ( n_p ) divides ( q ). ( n_q equiv 1 mod q ) and ( n_q ) divides ( p ).

Step 2: Analyzing ( n_p ) and ( n_q )

The next step is to analyze the possible values for ( n_p ) and ( n_q ) based on the Sylow Theorems.

For ( n_p ):

Since ( n_p ) divides ( q ) and ( q ) is a prime number, the possible values for ( n_p ) are ( 1 ) or ( q ).

If ( n_p 1 ), then there is a unique Sylow ( p )-subgroup which is normal in ( G ). For ( n_q ):

Since ( n_q ) divides ( p ) and ( p ) is also a prime, the possible values for ( n_q ) are ( 1 ) or ( p ).

If ( n_q 1 ), then there is a unique Sylow ( q )-subgroup which is also normal in ( G ).

Step 3: Case Analysis

Based on the analysis in Step 2, we can consider the following cases:

Case 1: ( n_p 1 )

Suppose ( n_p 1 ). Then ( G ) has a normal Sylow ( p )-subgroup ( P ) of order ( p ).

The subgroup ( P ) is cyclic since all groups of prime order are cyclic.

Case 2: ( n_q 1 )

Suppose ( n_q 1 ). Then ( G ) has a normal Sylow ( q )-subgroup ( Q ) of order ( q ).

The subgroup ( Q ) is also cyclic.

Case 3: ( n_p 1 ) and ( n_q 1 )

If both ( n_p 1 ) and ( n_q 1 ), then ( G ) has both a normal Sylow ( p )-subgroup ( P ) and a normal Sylow ( q )-subgroup ( Q ).

The group ( G ) can then be expressed as the semidirect product of these two normal subgroups. Since both ( P ) and ( Q ) are cyclic and their orders are coprime, ( G ) is cyclic.

Step 4: Conclusion

Any nontrivial subgroup of ( G ) must be a subgroup of either ( P ) or ( Q ) or the whole group itself.

Since both ( P ) and ( Q ) are cyclic, any nontrivial subgroup of ( G ) is cyclic.

Thus we conclude that any nontrivial subgroup of a group ( G ) of order ( pq ) must be cyclic.

Understanding the cyclic nature of subgroups in such groups is essential for deeper applications in group theory and related fields. Sylow's Theorems provide a robust framework for studying the structure of finite groups, especially those of smaller orders. This knowledge can be valuable in a variety of mathematical contexts, from pure mathematics to cryptography and beyond.