# Find an isomorphism between U_7 and Z_7

• kathrynag
In summary, the element in Z_{7} corresponding to \zeta^{m} for m=0,2,3,4,5, and 6 is 4^{m} in Z_{7}.
kathrynag

## Homework Statement

There is an isomorphism of $$U_{7}$$ with $$Z_{7}$$ in which $$\zeta$$=$$e^{(i2\pi}/7$$$$\leftrightarrow$$4. Find the element in $$Z_{7}$$ to which $$\zeta^{m}$$ must correspond for m=0,2,3,4,5, and 6.

## The Attempt at a Solution

$$\zeta^{0}$$=0
$$\zeta^{2}$$=4+$$_{7}$$4=1
$$\zeta^{3}$$=$$\zeta^{2}$$$$\zeta^{1}$$=4+$$_{7}$$1=-2

I don't know if I'm doing this right?

I would like to clarify a few things about your question. First, U_{7} and Z_{7} refer to the units and integers modulo 7, respectively. Isomorphism means that there exists a one-to-one mapping between the elements of these two groups that preserves their structure and operations. Therefore, the element \zeta in U_{7} corresponds to the integer 4 in Z_{7} in this specific isomorphism.

Now, for the element \zeta^{m}, we can use the property of isomorphism to find its corresponding element in Z_{7}. This property states that for any elements a and b in a group, their isomorphism will satisfy a^{m}\leftrightarrow b^{m}. So for m=0, we have \zeta^{0}=1 in U_{7}, which corresponds to 4^{0}=1 in Z_{7}. Similarly, for m=2, we have \zeta^{2}=1 in U_{7}, which corresponds to 4^{2}=2 in Z_{7}. For m=3, we have \zeta^{3}=\zeta^{2}\zeta^{1}=1\cdot\zeta=4 in U_{7}, which corresponds to 4^{3}=1 in Z_{7}. The same logic can be applied to find the corresponding elements for m=4,5, and 6.

I hope this clarifies your doubts and helps you solve the problem. Remember, isomorphism is a powerful tool that allows us to understand and compare different mathematical structures. Keep exploring and learning!

## 1. How do you define an isomorphism?

An isomorphism is a function that maps the elements of one mathematical structure to the elements of another structure in a way that preserves the structure and properties of both structures.

## 2. What is U_7 and Z_7?

U_7 is the set of positive integers less than 7 that are relatively prime to 7. Z_7 is the set of integers modulo 7, which includes the numbers 0, 1, 2, 3, 4, 5, and 6.

## 3. How do you find an isomorphism between U_7 and Z_7?

To find an isomorphism between U_7 and Z_7, we can use the function f(x) = x mod 7. This function maps the elements of U_7 to their corresponding elements in Z_7 and preserves the group structure.

## 4. Why is finding an isomorphism between U_7 and Z_7 important?

Finding an isomorphism between U_7 and Z_7 allows us to see the similarities between these two structures and understand their underlying mathematical properties. It also helps us solve problems in one structure by translating them into the other structure.

## 5. Are there other isomorphisms between U_7 and Z_7?

Yes, there are multiple isomorphisms between U_7 and Z_7. Another example is the function f(x) = 2^x mod 7, which also preserves the group structure and maps the elements of U_7 to their corresponding elements in Z_7.

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