Show that centralizerof an element is subgroup of H

To show that the centralizer of an element \( g \) in a group \( G \) is a subgroup, we follow the basic steps of subgroup verification. Let me walk you through the proof.

### Definition:
The **centralizer** of an element \( g \in G \), denoted \( C_G(g) \), is the set of all elements of \( G \) that commute with \( g \). In other words,
\[
C_G(g) = \{ h \in G \mid hg = gh \}.
\]
We need to show that \( C_G(g) \) is a subgroup of \( G \).

### Subgroup Criteria:
To show that \( C_G(g) \) is a subgroup, we verify the following three properties:
1. **Identity element**: The identity element of \( G \), denoted \( e \), must belong to \( C_G(g) \).
2. **Closure under multiplication**: If \( h_1, h_2 \in C_G(g) \), then their product \( h_1h_2 \) must also be in \( C_G(g) \).
3. **Closure under inverses**: If \( h \in C_G(g) \), then its inverse \( h^{-1} \) must also be in \( C_G(g) \).

### Proof:

#### 1. Identity Element:
The identity element \( e \in G \) satisfies the property \( eg = ge \) for any element \( g \in G \) (since \( e \) is the identity). Therefore, \( e \in C_G(g) \).

#### 2. Closure Under Multiplication:
Let \( h_1, h_2 \in C_G(g) \). By the definition of the centralizer, we know that:
\[
h_1g = gh_1 \quad \text{and} \quad h_2g = gh_2.
\]
Now, consider the product \( h_1h_2 \). We need to check if this product commutes with \( g \):
\[
(h_1h_2)g = h_1(h_2g) = h_1(gh_2) = (h_1g)h_2 = (gh_1)h_2 = g(h_1h_2).
\]
Since \( (h_1h_2)g = g(h_1h_2) \), we have \( h_1h_2 \in C_G(g) \).

#### 3. Closure Under Inverses:
Let \( h \in C_G(g) \). We know that \( hg = gh \). We need to show that the inverse \( h^{-1} \) also commutes with \( g \). Consider the following:
\[
h^{-1}g = h^{-1}(ghh^{-1}) = (h^{-1}h)gh^{-1} = egh^{-1} = gh^{-1}.
\]
Therefore, \( h^{-1}g = gh^{-1} \), so \( h^{-1} \in C_G(g) \).

### Conclusion:
Since \( C_G(g) \) contains the identity element, is closed under multiplication, and contains inverses, we conclude that \( C_G(g) \) is a subgroup of \( G \).

---

Would you like more details or have any questions?

Here are some related questions to expand on the concept:
1. What is the relationship between the centralizer and the center of a group?
2. How does the centralizer differ from the normalizer in a group?
3. Can the centralizer of an element ever be the entire group?
4. How is the centralizer of a subset of a group defined, and is it also a subgroup?
5. How do centralizers interact in the context of abelian groups?

**Tip**: Centralizers are important in studying group actions and conjugacy classes, especially in determining symmetries and simplifying group structures.

Show that the centralizer of an element is a subgroup of H.

Explore how to prove that the centralizer of an element in a group is a subgroup. This step-by-step solution delves into group theory concepts, including the centralizer, subgroup criteria, and basic group properties. Suitable for university-level students studying abstract algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation