# Algebra Question: Does R/Z under Addition Have Infinite Elements of Order 4?

• tgt
In summary, when considering R/Z under addition, there are an infinite number of elements of order 4. This can be shown by considering the cosets p/4+Z for primes p greater than 2, as there are an infinite number of primes. However, it can also be proven that all other examples fail to have an infinite number of elements of order 4.
tgt

## Homework Statement

Does R/Z under addition has an infinite number of elements of order 4? Where R denotes the real numbers and Z denotes the integers.

## The Attempt at a Solution

Yes. Consider the cosets p/4+Z for primes p greater than 2. Since there are an infinite number of primes, there are an infinite number of such cosets. But the answer says no for this question.

But each prime > 2 is 1 or 3 modulo 4. So if p is a prime, then the coset [p/4] = p/4 + Z is either equal to [1/4] or [3/4], isn't it?

Nice one.

Thanks. I'll leave it to you to prove that all other examples fail as well

## 1. What is R/Z under addition?

R/Z under addition refers to the quotient group formed by taking the real numbers (R) and dividing them by the integers (Z) under addition. This results in a group of equivalence classes, where each class represents a unique remainder when dividing a real number by an integer.

## 2. What does it mean for an element to have order 4?

The order of an element in a group refers to the smallest positive integer n for which the element raised to the power of n results in the identity element of the group. Therefore, an element of order 4 in R/Z under addition would mean that when this element is added to itself 4 times, the result is the identity element (0).

## 3. How do we know if R/Z under addition has infinite elements of order 4?

To determine if a group has infinite elements of a certain order, we need to find a pattern or general formula for the elements that have that particular order. In the case of R/Z under addition, we can use the fact that every real number has an infinite number of equivalent classes in this group. By considering these classes, we can find a pattern for elements of order 4 and conclude that there are infinite of them.

## 4. What is the significance of elements of order 4 in R/Z under addition?

Elements of order 4 in R/Z under addition are important because they provide a way to categorize and understand the structure of this group. By identifying these elements and studying their properties, we can gain insight into the behavior of the group as a whole. Additionally, elements of order 4 may have applications in other areas of mathematics and science.

## 5. Can you give an example of an element of order 4 in R/Z under addition?

Yes, an example of an element of order 4 in R/Z under addition is the equivalence class [0.25], which represents all real numbers that leave a remainder of 0.25 when divided by an integer. When this class is added to itself 4 times, the result is the identity element [0], making it an element of order 4. Other examples include [0.5], [0.75], [-0.75], etc.

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