Problem 4 - A Course In Group Theory

Question

Let \(G\) be any group and \(g\) be an element of \(G\). Prove directly that, \(C_{G}(g)=\\{x \in G: g x=x g\\}\) is a subgroup of \(G\).

Step-by-Step Solution

Answer

\(C_G(g)\) is a subgroup because it is non-empty and closed under group operations and inverses.

Step 1: Understand the Subgroup Criteria

To show that \(C_G(g)\) is a subgroup of \(G\), we need to verify the subgroup criteria: the set must be non-empty, closed under the group operation, and closed under taking inverses. This is because any subgroup must itself be a group.

Step 2: Show Non-emptiness

The identity element \(e\) of \(G\) satisfies \(g e = e g = g\) because \(e\) is the identity. Thus, \(e \in C_G(g)\), proving that \(C_G(g)\) is non-empty.

Step 3: Prove Closure Under Group Operation

Take any two elements \(x, y \in C_G(g)\). By definition, \(g x = x g\) and \(g y = y g\). We need to show \(x y \in C_G(g)\), i.e., \(g(xy) = (xy)g\). Using associativity, we have:\[g(xy) = (gx)y = (xg)y = x(gy) = x(yg) = (xy)g\].This shows \(x y \in C_G(g)\).

Step 4: Prove Closure Under Inverses

For each \(x \in C_G(g)\), we know \(g x = x g\). We want to prove that \(x^{-1} \in C_G(g)\), meaning \(g x^{-1} = x^{-1} g\). We take the inverse of the equation \(g x = x g\):- Multiply by \(x^{-1}\) from the right: \[g = x g x^{-1}\].- Multiply the above equation by \(g^{-1}\) from the left: \[g x^{-1} = x^{-1} g\].Thus, \(x^{-1} \in C_G(g)\).

Step 5: Conclusion

Since \(C_G(g)\) satisfies the criteria of being non-empty, closed under the group operation, and closed under taking inverses, it is confirmed that \(C_G(g)\) is a subgroup of \(G\).

Key Concepts

Subgroup Criteria

To determine if a set is a subgroup, it must satisfy three conditions: it must be non-empty, closed under the group operation, and closed under taking inverses. By verifying these properties, we ensure the set forms a group in itself.

• **Non-empty**: The identity element of the original group must be included.
• **Closure under the group operation**: If you take any two elements from the set and operate on them, the result should still be in the set.
• **Closure under inverses**: If an element is in the set, its inverse should also be part of the set.

These criteria help us confirm that our subgroup behaves like the larger group in terms of its structure.

Commutator Subgroup

The commutator subgroup is a specific type of subgroup that contains all possible commutators of a group. A commutator is formed by two elements, say \(a\) and \(b\), in the group and is expressed as \(a^{-1}b^{-1}ab\).
This subgroup encapsulates how far a group is from being abelian (commutative).

• If the commutator subgroup is trivial (just the identity), then the group is abelian.
• The commutator subgroup plays a crucial role in understanding the symmetry and structure of a group.

Exploring the commutator subgroup provides insights into the internal structure of the group.

Group Elements

Group elements are the building blocks of a group. These elements, when combined with the group operation, should adhere to group properties such as associativity, identity, and inversibility. Each group element has a unique inverse, fulfilling the basic group requirements.
In the context of our exercise, group elements are utilized to demonstrate properties such as:

• Commutation with a particular element \(g\).
• The existence of inverses which also satisfy commutation.

Understanding group elements helps us explore the dynamic interactions within the group.

Group Operation

The group operation is the binary operation that combines any two elements of a group to form another element within the same group. This operation must be associative, meaning the way in which elements are grouped together during the operation does not change the outcome.
In our example, closure under the group operation involves confirming that the operation on any two elements within our subset results in another element of the subset.

• The operation could be addition, multiplication, etc., depending on the group's nature.
• Associativity ensures that any rearrangement of the operation produces the same result.

The group operation is a central concept, allowing groups to maintain structure and coherence.