Proving the Abelian Property of a Group of Order 143 Using Sylow Theorems

Understanding the abelian property of a group is crucial in group theory. This involves checking whether a group ( G ) is commutative. A group ( G ) is abelian if and only if for all ( a, b in G ), ( ab ba ).

Introduction to Sylow Theorems

Sylow's theorems are a set of theorems in group theory that provide information about the existence and structure of subgroups whose order is a power of a prime number. They are particularly useful in analyzing the structure of groups of a specific order.

Factorization of 143 and Application to Sylow Theorems

A group ( G ) of order 143 can be factorized into prime factors as follows:

143 11 × 13

Here, both 11 and 13 are prime numbers. According to Sylow's theorems, we can analyze the number of Sylow ( p )-subgroups for each prime factor.

Determining the Number of Sylow ( p )-subgroups

Sylow 11-subgroups:

Let ( n_{11} ) be the number of Sylow 11-subgroups. By Sylow's theorems:

( n_{11} equiv 1 mod 11 ) ( n_{11} ) divides 13

Hence, the possible values for ( n_{11} ) are 1 or 13.

Sylow 13-subgroups:

Let ( n_{13} ) be the number of Sylow 13-subgroups. By Sylow's theorems:

( n_{13} equiv 1 mod 13 ) ( n_{13} ) divides 11

Hence, the possible values for ( n_{13} ) are 1 or 11.

Analysis of Cases

We now consider the different cases for ( n_{11} ) and ( n_{13} ).

Case 1: Both ( n_{11} 1 ) and ( n_{13} 1 )

There is a unique Sylow 11-subgroup ( P ), which is normal in ( G ). There is a unique Sylow 13-subgroup ( Q ), which is also normal in ( G ). The order of ( P ) is 11 and the order of ( Q ) is 13. Since ( P ) and ( Q ) are normal, ( G ) is the internal direct product of ( P ) and ( Q ):

( G cong P times Q )

Both ( P ) and ( Q ) are cyclic (since all groups of prime order are cyclic). The direct product of two cyclic groups is abelian. Therefore, ( G ) is abelian.

Case 2: Either ( n_{11} 13 ) or ( n_{13} 11 )

If either condition is true, the structure would become non-abelian, which contradicts the conditions imposed by Sylow's theorems.

The Sylow subgroups would not be normal, leading to contradictions in order and structure.

Conclusion

Given the conditions of Sylow's theorems, the only viable case is ( n_{11} 1 ) and ( n_{13} 1 ). This implies that both Sylow subgroups are normal. Therefore, ( G ) is abelian.

In conclusion, if ( G 143 ), then ( G ) is abelian.

Final Statement:

Thus, we have shown that if a group ( G ) has an order of 143, it must be abelian.