The set $\{-1, 1\}$ can be made into a group under multiplication, and the group $\mathbb{Z}_2$ is the group of integers modulo 2 under addition. Both groups are of order 2, so it is natural to look for an isomorphism between these two groups. An isomorphism is a bijective function that preserves the group operation.

To construct an isomorphism, let's define the map $\varphi: \{-1, 1\} \to \mathbb{Z}_2$ as follows:

$\varphi(-1) = 1 \quad \text{and} \quad \varphi(1) = 0$

Verifying the Isomorphism:

1. Preserving the operation:

   • In $\{-1, 1\}$, multiplication is the operation. So: $(-1) \times (-1) = 1 \quad \text{and} \quad (1) \times (1) = 1, \quad (-1) \times 1 = -1$ In $\mathbb{Z}_2$, addition modulo 2 is the operation. So: $1 + 1 = 0 \quad \text{and} \quad 0 + 0 = 0$

2. Check if the map preserves structure: We need to ensure that the map $\varphi$ preserves the operation between elements:

   • $\varphi((-1) \times (-1)) = \varphi(1) = 0$, and $\varphi(-1) + \varphi(-1) = 1 + 1 = 0$.
   • $\varphi(1 \times 1) = \varphi(1) = 0$, and $\varphi(1) + \varphi(1) = 0 + 0 = 0$.

3. Bijective: The map is clearly bijective because it is a one-to-one correspondence between the two elements.

Hence, this map is an isomorphism between $\{-1, 1\}$ and $\mathbb{Z}_2$, where multiplication in $\{-1, 1\}$ corresponds to addition modulo 2 in $\mathbb{Z}_2$.

Would you like further clarification on this concept, or do you have any other related questions?

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Here are 5 related questions:

1. How do isomorphisms preserve the group operation?
2. What is the identity element in $\{-1, 1\}$ and $\mathbb{Z}_2$?
3. Can you find other possible isomorphisms between two groups of order 2?
4. How do we verify that two groups are isomorphic?
5. What happens if the sets involved in an isomorphism are not of the same order?

Tip: For finite groups, a helpful strategy for checking if two groups are isomorphic is to match their elements one-to-one while ensuring the operation is preserved.