Problem 4 - A Book of Abstract Algebra

Question

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

Step-by-Step Solution

Answer

Thus, \(K\) is a subgroup of \(G\) because it satisfies identity, closure under multiplication, and closure under inverses.

Step 1: Definition and Requirements

To show that \(K\) is a subgroup of \(G\), we need to check two main conditions: (1) the identity element of \(G\) is in \(K\), and (2) \(K\) is closed under group operation and inverses. Here, \(K = \{x \in G: x^2 \in H\}.\)

Step 2: Identity Element

The identity element \(e\) of \(G\) satisfies \(e^2 = e \in H\), since \(H\) is a subgroup and contains the identity element of \(G\). Therefore, \(e \in K\).

Step 3: Closure Under Group Operation

Take any two elements \(a, b \in K\). By definition, we have \(a^2 \in H\) and \(b^2 \in H\). Since \(H\) is a subgroup, it is closed under multiplication, so \(a^2b^2 \in H\). Because \(G\) is abelian, \((ab)^2 = a^2b^2\). Thus, \((ab)^2 \in H\), and so \(ab \in K\).

Step 4: Closure Under Inverses

Take any element \(a \in K\). By definition, \(a^2 \in H\). Since \(H\) is a subgroup, it is closed under inverses, so \((a^2)^{-1} = a^{-2} = (a^{-1})^2 \in H\). Thus, \(a^{-1} \in K\).

Step 5: Conclusion

Since \(K\) satisfies the conditions for identity, closure under group operation, and closure under inverses, \(K\) is a subgroup of \(G\).

Key Concepts

Subgroup Criteria

To determine whether a subset forms a subgroup within a group, several criteria must be met. These conditions ensure that the subset maintains the group structure. Here are the three essential criteria for identifying a subgroup:

• Contains the identity element: The subset must include the identity element of the original group, ensuring a neutral starting point for operations.
• Closed under group operation: Performing the group operation on any two elements from the subset should yield a result also contained within the subset.
• Closed under inverses: For each element in the subset, its inverse (according to the group's operation) should also be present within the subset.

These criteria guarantee that the subgroup has the same structural properties as the main group, allowing any operations within the subgroup to behave consistently with those in the group.

Group Operation

In group theory, the concept of group operation is fundamental. A group operation is a binary operation that combines any two elements from the group to yield another element within the same group. This operation must adhere to specific rules to maintain group structure:

• Associativity: The order of applying the operation does not change the outcome. For example, for any elements \(a, b, c\), we have \((a \, \text{op} \, b) \, \text{op} \, c = a \, \text{op} \, (b \, \text{op} \, c)\).
• Identity Element: There exists an element in the group, known as the identity element, which when used in the operation with any element from the group, results in that same element.
• Inverse Elements: For each element in the group, there exists an inverse element such that when the operation is performed on these two elements, the result is the identity element.

Understanding group operation helps in exploring properties like closure and inverses, which are essential for proving if subsets like \(K\) are subgroups.

Closure Properties

In mathematics, closure properties are crucial when discussing subgroups. The closure of a set under some operation means that performing the operation on elements from the set always results in an element still within the set.For a subgroup, closure under the group operation ensures that combining any two elements from the subgroup through the defined operation keeps you within the subgroup. Similarly, closure under inverses ensures that the inverse of any element in the subgroup also resides within the subgroup.In our context of proving \(K\) is a subgroup, demonstrating that for elements \(a, b \in K\), the element \(ab\) is also in \(K\), illustrates closure under the group operation. Meanwhile, proving \(a^{-1} \in K\) verifies closure under inverses. Thus, closure properties help solidify the argument that these elements form a coherent group under the subset's operations.

Identity Element

The identity element is a fundamental concept in group theory. It acts as the neutral element for the group operation, meaning when you combine it with any other element in the group, that element remains unchanged.In any group \(G\), the identity element is denoted as \(e\). For a subset like \(K\) to be a subgroup of \(G\), it must include this identity element. In our exercise, since \(G\) itself is a group, it includes \(e\) such that \(e^2 = e\), and by the property of subgroup \(H\), where \(e \in H\), it follows that \(e \in K\). This inclusion is crucial for validating \(K\) as a subgroup because it establishes a base element that ensures consistency and coherence of all operations within \(K\).

Inverse Elements

Inverse elements are quintessential in the structure of groups, emphasizing the property that for each element in a group, there's a counterpart that "undoes" the operation to return to the identity element.For any element \(a\) in a group \(G\), there exists an inverse element \(a^{-1}\) such that the operation between them results in the identity: \(a \, \text{op} \, a^{-1} = e\), where \(e\) is the identity element. In the context of proving \(K\) is a subgroup, if \(a \in K\), then \(a^2 \in H\). Since \(H\) is a subgroup, it's closed under inverses. Thus, \((a^2)^{-1} = (a^{-1})^2 \in H\), ensuring \(a^{-1} \in K\). Including inverse elements solidifies \(K\)'s standing as a subgroup, ensuring any operation can revert within \(K\), mimicking the parent group's reversible structure.