Problem 1 - A Book of Abstract Algebra

Question

Let \(G\) be an abelian group. Let \(H=\left\\{x^{2}: x \in G\right\\}\) and \(K=\left\\{x \in G: x^{2}=e\right\\}\). Prove that \(f(x)=x^{2}\) is a homomorphism of \(G\) onto \(H\).

Step-by-Step Solution

Answer

The function \(f(x) = x^2\) is a homomorphism of \(G\) onto \(H\) because it preserves the group operation and maps every element in \(H\).

Step 1: Define Group Homomorphism

To show that the function \(f(x) = x^2\) is a homomorphism, we need to verify that it preserves the group operation. Specifically, for all elements \(x, y \in G\), the function should satisfy \(f(xy) = f(x)f(y)\).

Step 2: Verify Homomorphism Property

Evaluate \(f(xy)\): \(f(xy) = (xy)^2\). By using the property of the group \(G\) being abelian (commutative), we have \((xy)^2 = xyxy = x^2y^2\), which is equal to \(f(x)f(y)\). Thus, \(f(xy) = f(x)f(y)\) is satisfied, confirming \(f\) is a homomorphism.

Step 3: Prove Onto Function

To prove that \(f\) is onto, for each element \(h \in H\), there must exist an \(x \in G\) such that \(f(x) = h\). By definition, every element \(h \in H\) can be written as \(x^2\) for some \(x \in G\) (since \(H = \{x^2: x \in G\}\)). Therefore, \(f(x) = x^2 = h\), showing that \(f\) is onto.

Key Concepts

Abelian Group

Abelian groups are a fundamental concept in group theory. These groups are characterized by the fact that their group operation is commutative. This means that for any two elements, say \(a\) and \(b\), in the group \(G\), the operation satisfies \(a \cdot b = b \cdot a\). This property simplifies many mathematical proofs and computations.

An abelian group has the following properties:

• Closure: For any elements \(a\) and \(b\) in the group, the result of the operation \(a \cdot b\) is also an element of the same group.
• Associativity: The group operation is associative. So for elements \(a\), \(b\), and \(c\), \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
• Identity Element: There exists an element, often denoted by \(e\), such that for all elements \(a\), \(e \cdot a = a \cdot e = a\).
• Inverse Element: For each element \(a\), there is an element \(b\) such that \(a \cdot b = b \cdot a = e\).
• Commutative Property: As mentioned, \(a \cdot b = b \cdot a\) for any \(a\), \(b\) in the group.

Abelian groups are named after the mathematician Niels Henrik Abel, and they play a crucial role in various areas of mathematics, including algebra and number theory.

Group Homomorphism

A group homomorphism is a map between two groups that preserves the structure of the groups. This means that when the group operation is applied to any two elements in the first group, the result of mapping them is equivalent to mapping each and then applying the operation in the second group.

For a group homomorphism \(f: G \to H\), where both \(G\) and \(H\) are groups, the following condition must be satisfied:

• Operation Preservation: For all \(x, y \in G\), \(f(xy) = f(x)f(y)\).

In the context of the exercise, the function \(f(x) = x^2\) must be shown to preserve the group operation in \(G\), making it a homomorphism. If \(G\) is an abelian group, this is simpler because the commutative property helps show that \((xy)^2 = x^2y^2\), which means the function preserves the group's structure. This means it maps group elements in \(G\) into elements in \(H\), preserving the defined operation.

Commutative Property

The commutative property is crucial in the study of abelian groups and plays a significant role in proving many properties within group theory. This property states that if a group is commutative, also known as abelian, then for any two elements \(a\) and \(b\) in the group, swapping them in any operation does not change the result.

More formally, for a group \(G\), for any elements \(a, b \in G\), the equations \(a \cdot b = b \cdot a\) hold true. This property simplifies the analysis of the structures and transformations within the group. Because of its simplicity, it is one of the defining features of abelian groups.

Here's why the commutative property is important in proving homomorphism, as detailed in the exercise:

• When verifying that a function \(f\) is a homomorphism for an abelian group, the commutative property allows the simplification of expressions, which makes it straightforward to verify that the function preserves group operations.
• It also ensures that any transformations or mappings of elements retain their natural order, as stipulated by the group's operation.

The commutative property is easy to overlook, but it plays a key role in group theory, aiding in proving many theorems and propositions.