) Define Kernel of homomorphism. Let (G, o) and ( G ^ * ,o^ * ) be two groups, and f / G G^ * be a homomorphism then show that Ker

() is a normal of (G, o)

### Kernel of a Homomorphism

Let \((G, \circ)\) and \((G^*, \circ^*)\) be two groups, and let \(f: G \to G^*\) be a homomorphism. The **kernel** of the homomorphism \(f\), denoted by \(\text{Ker}(f)\), is defined as:

\[
\text{Ker}(f) = \{g \in G \mid f(g) = e^*\}
\]

where \(e^*\) is the identity element in the group \(G^*\).

### Proof: \(\text{Ker}(f)\) is a Normal Subgroup of \((G, \circ)\)

To show that \(\text{Ker}(f)\) is a normal subgroup of \((G, \circ)\), we need to establish two things:

1. \(\text{Ker}(f)\) is a **subgroup** of \(G\).
2. \(\text{Ker}(f)\) is **normal** in \(G\), i.e., for all \(g \in G\) and \(k \in \text{Ker}(f)\), \(g \circ k \circ g^{-1} \in \text{Ker}(f)\).

#### 1. **\(\text{Ker}(f)\) is a Subgroup of \(G\)**

To prove that \(\text{Ker}(f)\) is a subgroup, we check the following three conditions:

- **Identity**: The identity element \(e\) of \(G\) must be in \(\text{Ker}(f)\).
  
  Since \(f\) is a homomorphism, \(f(e) = e^*\). Therefore, \(e \in \text{Ker}(f)\).

- **Closure**: If \(k_1, k_2 \in \text{Ker}(f)\), then \(k_1 \circ k_2 \in \text{Ker}(f)\).
  
  For \(k_1, k_2 \in \text{Ker}(f)\), we have:
  \[
  f(k_1) = e^* \quad \text{and} \quad f(k_2) = e^*
  \]
  Since \(f\) is a homomorphism:
  \[
  f(k_1 \circ k_2) = f(k_1) \circ^* f(k_2) = e^* \circ^* e^* = e^*
  \]
  Thus, \(k_1 \circ k_2 \in \text{Ker}(f)\).

- **Inverses**: If \(k \in \text{Ker}(f)\), then \(k^{-1} \in \text{Ker}(f)\).
  
  For \(k \in \text{Ker}(f)\), we have \(f(k) = e^*\). Since \(f\) is a homomorphism:
  \[
  f(k^{-1}) = f(k)^{-1} = (e^*)^{-1} = e^*
  \]
  Therefore, \(k^{-1} \in \text{Ker}(f)\).

Since \(\text{Ker}(f)\) satisfies identity, closure, and inverses, it is a subgroup of \(G\).

#### 2. **\(\text{Ker}(f)\) is Normal in \(G\)**

To show that \(\text{Ker}(f)\) is normal, we need to prove that for all \(g \in G\) and \(k \in \text{Ker}(f)\):
\[
g \circ k \circ g^{-1} \in \text{Ker}(f)
\]

Let \(g \in G\) and \(k \in \text{Ker}(f)\). Then:
\[
f(k) = e^*
\]
We want to show that:
\[
f(g \circ k \circ g^{-1}) = e^*
\]
Using the homomorphism property:
\[
f(g \circ k \circ g^{-1}) = f(g) \circ^* f(k) \circ^* f(g^{-1})
\]
Since \(f(k) = e^*\), we have:
\[
f(g \circ k \circ g^{-1}) = f(g) \circ^* e^* \circ^* f(g^{-1}) = f(g) \circ^* f(g^{-1})
\]
Because \(f\) is a homomorphism:
\[
f(g) \circ^* f(g^{-1}) = f(g \circ g^{-1}) = f(e) = e^*
\]
Thus:
\[
f(g \circ k \circ g^{-1}) = e^*
\]
This shows that \(g \circ k \circ g^{-1} \in \text{Ker}(f)\).

Since \(g \circ k \circ g^{-1} \in \text{Ker}(f)\) for all \(g \in G\) and \(k \in \text{Ker}(f)\), \(\text{Ker}(f)\) is a normal subgroup of \(G\).

### Conclusion
The kernel of a homomorphism \(f: G \to G^*\), \(\text{Ker}(f)\), is a normal subgroup of the group \(G\).

Would you like further details on this proof or have any additional questions?

### Follow-Up Questions:
1. What is the significance of the kernel in determining whether a homomorphism is injective?
2. How does the first isomorphism theorem relate to the kernel of a homomorphism?
3. Can the kernel of a homomorphism ever be the entire group \(G\)?
4. How would you determine the kernel of a specific homomorphism \(f: G \to G^*\) given an example?
5. What role does the normality of the kernel play in quotient groups?

### Tip:
When analyzing homomorphisms, always start by identifying the kernel, as it provides crucial information about the nature of the homomorphism, such as injectivity and the structure of quotient groups.

Explore the definition and properties of the kernel of a homomorphism between groups. Understand how the kernel is a normal subgroup of the group G, defined by the homomorphism f: G → G*. Learn through a detailed proof and examples. Suitable for advanced undergraduate students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation