Introduction to Finding Generators of ( U_{50} )

Understanding the generators of ( U_{50} ), the group of units modulo 50, is a fundamental task in number theory and group theory. In this article, we will walk through the step-by-step process to identify these generators. This guide will help you grasp the concept and provide a detailed explanation of each step.

Understanding ( U_{50} )

( U_{50} ) is the group of units modulo 50. It consists of all integers less than 50 that are coprime to 50. That is, the integers ( k ) such that ( 1 leq k gcd(k, 50) 1.

Step 1: Identifying the Elements of ( U_{50} )

The prime factorization of 50 is 50 2 × 5^2. Hence, any integer not divisible by 2 or 5 will be in ( U_{50} ). The elements of ( U_{50} ) are as follows:

1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49

Therefore, the set ( U_{50} ) can be written as:

( U_{50} {1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49} )

Step 2: Determining the Order of ( U_{50} )

The order of the group ( U_{50} ) is given by Euler's totient function ( phi(50) ).

( phi(50) 50 left(1 - frac{1}{2}right) left(1 - frac{1}{5}right) 50 times frac{1}{2} times frac{4}{5} 20 )

Hence, the order of ( U_{50} ) is 20.

Step 3: Finding the Generators of ( U_{50} )

A generator ( g ) of ( U_{50} ) must have an order equal to ( phi(50) 20 ). To find these generators, we need to check the order of each element in ( U_{50} ).

The possible orders of elements in ( U_{50} ) must divide 20. These are 1, 2, 4, 5, 10, and 20. We will check each element to see if it has an order of 20.

Example: Checking the Generator 3

Let's verify that 3 has an order of 20.

( 3^1 equiv 3 mod 50 ) ( 3^2 equiv 9 mod 50 ) ( 3^4 equiv 81 equiv 31 mod 50 ) ( 3^5 equiv 243 equiv 43 mod 50 ) ( 3^{10} equiv 43^2 equiv 1849 equiv 49 mod 50 ) ( 3^{20} equiv (3^{10})^2 equiv 49^2 equiv 2401 equiv 1 mod 50 )

Since ( 3^{20} equiv 1 mod 50 ) and no smaller power of 3 is congruent to 1 modulo 50, 3 is a generator.

Step 4: Listing the Generators

After checking each element, the generators of ( U_{50} ) are:

3, 7, 9, 11, 17, 19, 29, 31, 37, 43

These elements have an order of 20 and can generate all other elements of ( U_{50} ) through their powers.

Conclusion

Identifying the generators of ( U_{50} ) involves determining the elements of ( U_{50} ), calculating the order of each element, and verifying which elements have an order of 20. In this example, we found that 3, 7, 9, 11, 17, 19, 29, 31, 37, and 43 are the generators of ( U_{50} ).

If you need further assistance or specific calculations for any element, feel free to reach out. Understanding these steps will not only help in solving problems related to ( U_{50} ) but also in grasping the broader concepts of group theory.