To show that the groups G = Z and H = 3Z are isomorphic, we need to find an isomorphism, which is a bijective function that preserves the group operation.

1. Define the function f: G - H by f(x) = 3x for all x in G. This function is clearly a bijection (one-to-one and onto).

2. We need to show that f is operation-preserving. In other words, for all a, b in G, we need to show that f(a + b) = f(a) + f(b).

Let's take arbitrary a, b in G. Then f(a + b) = 3(a + b) = 3a + 3b = f(a) + f(b).

So, f is an isomorphism from G to H. Therefore, G and H are isomorphic.

Question

To show that the groups G = Z and H = 3Z are isomorphic, we need to find an isomorphism, which is a bijective function that preserves the group operation.

1. Define the function f: G -> H by f(x) = 3x for all x in G. This function is clearly a bijection (one-to-one and onto).

2. We need to show that f is operation-preserving. In other words, for all a, b in G, we need to show that f(a + b) = f(a) + f(b).

Let's take arbitrary a, b in G. Then f(a + b) = 3(a + b) = 3a + 3b = f(a) + f(b).

So, f is an isomorphism from G to H. Therefore, G and H are isomorphic.

Knowee AI · Accepted Answer

To show that the groups G = Z and H = 3Z are isomorphic, we need to find an isomorphism, which is a bijective function that preserves the group operation.

1. Define the function f: G -> H by f(x) = 3x for all x in G. This function is clearly a bijection (one-to-one and onto).

2. We need to show that f is operation-preserving. In other words, for all a, b in G, we need to show that f(a + b) = f(a) + f(b).

Let's take arbitrary a, b in G. Then f(a + b) = 3(a + b) = 3a + 3b = f(a) + f(b).

So, f is an isomorphism from G to H. Therefore, G and H are isomorphic.

Show that the groups G = Z and H = 3Z are isomorphic.

Question

Solution

Similar Questions

Upgrade your grade with Knowee