Problem 5 - A Book of Abstract Algebra

Question

Let \(G\) be a finite group, say a group with \(n\) elements, and let \(S\) be a nonempty subset of \(G\). Suppose \(e \in S\), and that \(S\) is closed with respect to multiplication. Prove that \(S\) is a subgroup of \(G\). (HINT: It remains to prove that \(G\) is closed with respect to inverses. Let \(G=\left\\{a_{1}, \ldots, a_{n}\right\\} ;\) one of these elements is \(e .\) If \(a_{i} \in G\), consider the distinct elements \(\left.a_{i} a_{1}, a_{i} a_{2}, \ldots, a_{i} a_{n},\right\),

Step-by-Step Solution

Answer

The subset \( S \) is a subgroup of \( G \) since it contains the identity, is closed under multiplication, and closed under inverses.

Step 1: Define a Group and Subgroup

A group is a set paired with an operation that satisfies closure, associativity, has an identity element, and each element has an inverse. A subgroup is a subset that itself is a group with respect to the same operation.

Step 2: Verify Identity Element in Subset

We are given that the identity element \( e \) is in the subset \( S \). This is one of the criteria for \( S \) being a subgroup.

Step 3: Verify Closure Under Multiplication

The subset \( S \) is closed under multiplication by the problem's statement, meaning for any \( a, b \in S \), the product \( ab \in S \). This verifies one of the subgroup criteria.

Step 4: Verify Closure Under Inverses

To prove closure under inverses, select any element \( a \in S \). We must show \( a^{-1} \in S \). Since \( a \) and the identity \( e \) are in \( S \), if \( S \) includes multiplications like \( aa^{-1} = e \), then \( a^{-1} \) must also be in \( S \) to maintain closure.

Step 5: Apply Properties of Finite Groups

In a finite group \( G \) with the subset \( S \), if adding \( a^{-1} \) ensures \( a^{-1} \in S \), \( S \) is confirmed to have all necessary group properties.

Step 6: Conclude Subgroup Properties

Thus, \( S \) contains the identity, is closed under the operation, and includes inverses, meeting all subgroup criteria.

Key Concepts

Finite Group

In the study of group theory, a finite group is one with a finite number of elements. So, when we talk about a group that has "n elements," we are discussing a collection that you can list out completely. The set is paired with a specific operation, like addition or multiplication, which forms a complete structure known as a group.

Key characteristics of any group include:

• Closure: Performing the group operation on any two elements in the group returns another element from the group.
• Associativity: When applying the operation, the grouping of elements does not affect the result.
• Identity Element: There exists an element within the group that does not change other elements when used in the operation.
• Inverses: Every element has another element that combines with it to return the identity element.

A finite group holds all these properties, just with a limited number of elements. Understanding finite groups helps in diving deeper into concepts like subgroups.

Subgroup

A subgroup is like a mini-group within a larger group. It is a subset that itself satisfies all the properties required to be a group under the same operation as the larger group.

For a subset to qualify as a subgroup, it must be closed under the group operation. It should also contain the identity element of the larger group and have inverses for each of its elements. In addition:

• Closure: If you take any two elements from the subset and apply the group operation, the result should be an element in the subset itself.
• Identity: It must include the identity element from the parent group.
• Inverses: Every element of the subset must have an inverse within that same subset.

Essentially, a subgroup inherits the characteristics of the larger group but exists as its independent group.

Closure Property

The closure property is essential in group theory as it ensures consistency when performing operations within the group or any of its subgroups.

When we say a set is closed under a particular operation, it means that taking any elements from that set, applying the operation, and arriving at an outcome should yield another element still within that set. For example, if you multiply any two numbers from a set, and the result is still a number from the same set, then the set is closed under multiplication.

For a subset to be a subgroup, demonstrating closure under its operation is crucial. For instance, in our exercise, if any two elements from the subset \( S \) multiply to give another element in \( S \), \( S \) is closed under multiplication. This property is fundamental to establishing that \( S \) could be a subgroup of \( G \).

Identity Element

The identity element is a cornerstone concept in group theory. It is the element in a group that, when used in the group's operation with any other element, leaves the other element unchanged.

For example, in a group based on addition, the identity is 0 because adding 0 to any number does not change the number. If the group's operation is multiplication, the identity is 1, since multiplying any number by 1 results in the original number.

In terms of forming a subgroup, the identity element from the larger group must also appear in the subset. This inclusion helps to ensure that, within the subset, the operation can result in an outcome that uses the identity, maintaining the subgroup's closure. Including the identity element satisfies one of the critical conditions for a set to be considered a valid subgroup.