Question

In the following, verify if the mappings defined are homomorphisms, and in those cases in which they are homomorphisms, determine the kernel

G is the group of nonzero real numbers under multiplication, G = G, (x) = x² ah x e g

Answer 1

To verify if a mapping defined on the group of nonzero real numbers under multiplication is a homomorphism, we need to show that φ(ab) = φ(a)φ(b) for all a, b in G. The kernel of the mapping is the set of elements in G that are mapped to the identity element, which in this case is 1.

Step-by-step explanation:

The question asks us to verify if a mapping defined on the group of nonzero real numbers under multiplication is a homomorphism, and if so, to determine its kernel.

Let's consider the mapping φ: G -> G defined by φ(x) = x², where x is an element of G.

To show that φ is a homomorphism, we need to prove that φ(ab) = φ(a)φ(b) for all a, b in G.

For example, let's take a = 2 and b = 3.

φ(2 * 3) = φ(6) = 6² = 36, and φ(2)φ(3) = 4 * 9 = 36.

Since φ(ab) = φ(a)φ(b) for any a, b in G, we can conclude that φ is a homomorphism.

The kernel of φ is the set of elements in G that are mapped to the identity element of G under φ. In this case, the identity element is 1.

To find the kernel, we set φ(x) = 1 and solve for x:

x² = 1.

The solutions to this equation are x = 1 and x = -1.

Therefore, the kernel of φ is {1, -1}.