# Show that Z_12^* and Z_8^* are isomorphic groups

• Mr Davis 97
In summary, to show that ##\mathbb{Z}_8^*## and ##\mathbb{Z}_12^*## are isomorphic, we can use a function that maps the elements of one group to the other, such as 1 to 1, 3 to 5, 5 to 7, and 7 to 11. This function is injective and surjective, and the homomorphism property can be proven by showing it is satisfied for each combination from the domain. However, a quicker way to show isomorphism is to note that there are only two abelian groups of order 4, the cyclic group C_4 and the Klein group C_2 \times C_
Mr Davis 97

## Homework Statement

Show that ##\mathbb{Z}_8^*## and ##\mathbb{Z}_12^*## are isomorphic, where ##\mathbb{Z}_n^* = \{x \in \mathbb{Z} ~|~ \exists a \in \mathbb{Z}_n(ax \equiv 1~(mod~n)) \}##, and the group operation is regular multiplication.

## The Attempt at a Solution

We can see that ##\mathbb{Z}_8^* = \{1,3,5,7 \}## and ##\mathbb{Z}_8^* = \{1,5,7,11 \}##

The only possible isomorphism I can think if is a function that maps from the former to the latter such that 1 goes to 1, 3 to 5, 5 to 7, and 7 to 11. The function is obviously injective and surjective. Is the only way to show that this satisfies the homomorphism property to show that it is satisfied for each combination from the domain? This would seem to be a tedious process.

The quick way is to note that up to isomorphism there are two abelian groups of order 4, the cyclic group $C_4$ (which contains an element of order 4) and the klein group $C_2 \times C_2$ (which does not contain an element of order 4).

## 1. What does it mean for two groups to be isomorphic?

Two groups are isomorphic if there exists a bijective function between the elements of the two groups that preserves the group operation. In other words, the two groups have the same underlying structure.

## 2. What are Z_12^* and Z_8^*?

Z_12^* and Z_8^* are both groups under multiplication modulo 12 and 8 respectively. This means that the elements of the groups are the integers from 1 to 11 for Z_12^* and from 1 to 7 for Z_8^*, and the group operation is multiplication followed by taking the remainder when divided by 12 or 8.

## 3. How can I show that Z_12^* and Z_8^* are isomorphic?

To show that two groups are isomorphic, you need to find a bijective function between the elements of the two groups that preserves the group operation. In this case, you can show that the function f: Z_12^* -> Z_8^* defined as f(x) = x^2 mod 8 is bijective and preserves the group operation, thus proving the isomorphism.

## 4. Why is it important to show that two groups are isomorphic?

Showing that two groups are isomorphic provides a deeper understanding of the structures of these groups and allows for the transfer of knowledge and results between the two groups. It also enables us to solve problems in one group by using techniques and results from the other group.

## 5. Are there any other examples of isomorphic groups?

Yes, there are many examples of isomorphic groups, such as the group of even integers under addition and the group of rational numbers under multiplication. It is a common concept in abstract algebra and has many applications in various fields of mathematics.

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