Show the Units of Zn with modular multiplication are a group

In summary, the conversation discusses the task of proving that the set of all elements of $\Bbb{Z}_n$ that are coprime with $n$ forms a group under modular addition. The speaker has already shown associativity, identity, and closure, but is struggling to prove the existence of an inverse. The conversation suggests using the fact that $gcd(r,n) = 1$ if and only if there exist integers $s$ and $t$ such that $rs + nt = 1$, and to take this equation mod $n$.
  • #1
E01
8
0
I am trying to do an exercise where I am showing that the set of all elements of $\Bbb{Z}_n$ that are coprime with n form a group under modular addition.

So far I have shown associativity, identity, and closure, but I'm having trouble showing the existence of an inverse. I know I can't use reciprocals and I can't find a way to prove that for $r \in U_n$ there exists some $t \in U_n$ such that $tr$ has a remainder of 1 when divided by n.

Any hints?
 
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  • #2
E01 said:
I am trying to do an exercise where I am showing that the set of all elements of $\Bbb{Z}_n$ that are coprime with n form a group under modular addition.

So far I have shown associativity, identity, and closure, but I'm having trouble showing the existence of an inverse. I know I can't use reciprocals and I can't find a way to prove that for $r \in U_n$ there exists some $t \in U_n$ such that $tr$ has a remainder of 1 when divided by n.

Any hints?

Hi E01,

Use the fact that $gcd(r,n) = 1$ if and only if there exist integers $s$ and $t$ such that $rs + nt = 1$.
 
  • #3
E01 said:
I am trying to do an exercise where I am showing that the set of all elements of $\Bbb{Z}_n$ that are coprime with n form a group under modular addition.

So far I have shown associativity, identity, and closure, but I'm having trouble showing the existence of an inverse. I know I can't use reciprocals and I can't find a way to prove that for $r \in U_n$ there exists some $t \in U_n$ such that $tr$ has a remainder of 1 when divided by n.

Any hints?

Hint #2: take Euge's equation mod $n$.
 

Related to Show the Units of Zn with modular multiplication are a group

1. What is a group in modular multiplication?

A group in modular multiplication refers to a set of elements that follow a specific mathematical structure. In this case, the group consists of all the units of Zn (integers that have a multiplicative inverse modulo n) under the operation of modular multiplication.

2. Why is it important to show that the units of Zn with modular multiplication form a group?

It is important to show that the units of Zn with modular multiplication form a group because it proves that the set has a well-defined structure and follows certain properties. This can help in solving various mathematical problems and understanding the behavior of integers under modular arithmetic.

3. How is the identity element defined in the group of units of Zn with modular multiplication?

The identity element in the group of units of Zn with modular multiplication is defined as the integer 1, since it is the only number that, when multiplied with any other unit of Zn, results in the same number. In other words, 1 is the multiplicative identity in this group.

4. Can you explain the closure property in the group of units of Zn with modular multiplication?

The closure property in the group of units of Zn with modular multiplication means that when two units of Zn are multiplied together, the result is also a unit of Zn. This is because the set only contains integers that have a multiplicative inverse modulo n, so any combination of these integers will also have a multiplicative inverse modulo n.

5. How does the inverse property work in the group of units of Zn with modular multiplication?

The inverse property in the group of units of Zn with modular multiplication states that for every unit a in Zn, there exists another unit b such that a * b ≡ 1 (mod n). This means that every element in the group has an inverse element, which is necessary for the group to have a well-defined structure and for all operations to be reversible.

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