Is the Units Group of Z Modulo 34 Cyclic?

In summary, the conversation discusses the question of whether the group (ℤ/34/ℤ)x is cyclic. It is mentioned that the theorem stating that any units group of Z modulo n is cyclic if and only if n = 1, 2, 4, pk, and 2pk (where p is an odd prime) has not been discussed in class. However, thanks to Deveno, it is known that this theorem applies in this case with p = 17. The question is then raised if there is another way to prove the group is cyclic without brute force. It is clarified that p does not necessarily have to be equal to 4k+3 for the theorem to hold. It is suggested that
  • #1
Bachelier
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I came across this question. Is the group (ℤ/34/ℤ)x cyclic?
We haven't discussed the theorem in class that any units group of Z modulo n is iff n = 1, 2, 4, pk and 2pk (where p is an odd prime). But thanks to Deveno I know about it. So in this case, p =17 works, so the group is cyclic?

but is there a different way to show it besides using brute force .
 
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  • #2


BTW, p doesn't have to be equal to 4k+3 for this to be true, correct?
 
Last edited:
  • #3


I guess I can show that ℤ/34/ℤx is ≈ to ℤ/17/ℤx and by the field's finite subgroup theorem, it is a cyclic.
 

FAQ: Is the Units Group of Z Modulo 34 Cyclic?

What is a unit group of Z modulo?

The unit group of Z modulo (denoted as (Zn)*) is defined as the set of all elements in the integers modulo n that have a multiplicative inverse. In other words, these are the elements in Zn that have a number that, when multiplied with them, results in a remainder of 1 when divided by n.

How is the unit group of Z modulo calculated?

The unit group of Z modulo is calculated using the Euler's totient function, φ(n). This function counts the number of positive integers less than n that are relatively prime to n. The unit group of Z modulo is then given by (Zn)* = {a ∈ Zn | gcd(a, n) = 1}.

What is the order of the unit group of Z modulo?

The order of the unit group of Z modulo, also known as the size of (Zn)*, is given by φ(n), which is the number of elements in the group. This is because the unit group only contains elements that have a multiplicative inverse, and the rest of the elements in Zn do not have a multiplicative inverse.

What is the significance of the unit group of Z modulo?

The unit group of Z modulo has several applications in number theory and cryptography. It is used in various algorithms, such as the RSA algorithm, for secure data encryption and decryption. It is also used in the Chinese remainder theorem, which allows for efficient computations involving large numbers.

How is the unit group of Z modulo related to cyclic groups?

The unit group of Z modulo is a cyclic group, meaning that it can be generated by a single element. This element is called a primitive root, and it has the property that its powers cover all the elements in the group. In other words, every element in the unit group of Z modulo can be expressed as a power of the primitive root. This property makes the unit group of Z modulo useful in various applications, such as in the Diffie-Hellman key exchange protocol.

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