Problem 21 - A First Course in Abstract Algebra

Question

A student is asked to show that if \(H\) is a normal subgroup of an abelian group \(G\), then \(G / H\) is abelian. The student's proof starts as follows: We must show that \(G / H\) is abelian. Let \(a\) and \(b\) be two elements of \(G / H\). a. Why does the instructor reading this proof expect to find nonsense from here on in the student's paper? b. What should the student have written? c. Complete the proof.

Step-by-Step Solution

Answer

The instructor expects an error because the student treats cosets as elements. Each coset in \(G/H\), not individuals from \(G\), should be noted. When done correctly, the proof concludes \(G/H\) is abelian.

Step 1: Understanding Normal Subgroups

Recall that a subgroup \( H \) of a group \( G \) is called normal if for every element \( g \in G \), the conjugate \( gHg^{-1} \) is equal to \( H \). Notationally, this is expressed as \( H rianglelefteq G \).

Step 2: Recognizing the Structure of an Abelian Group

An abelian group is a group where for any two elements \( x, y \in G \), the equation \( xy = yx \) holds. This means the group operation is commutative.

Step 3: Identifying the Error in the Proof's Assumption

The instructor notices a problem because in the student's paper, "elements of \( G / H \)" should be cosets, not individual elements from \( G \). Consequently, if \( a, b \) are considered elements of \( G / H \), they should be denoted as cosets like \( aH \) and \( bH \), not just elements.

Step 4: Correcting the Initial Assumption

The student should have written: "Let \( aH \) and \( bH \) be two elements of \( G / H \), where \( a, b \in G \)." This specifies that \( aH \) and \( bH \) are cosets, which are the actual elements of the quotient group \( G / H \).

Step 5: Completing the Proof

Since \( G \) is abelian, for any \( a, b \in G \), \( ab = ba \). Thus, the product of two cosets \( aH \) and \( bH \) in \( G / H \) is defined as \((aH)(bH) = (ab)H\). Because \( ab = ba \), we have \[(ab)H = (ba)H = (bH)(aH).\] This shows commutativity, hence \( G / H \) is abelian.

Key Concepts

Normal Subgroup

In the realm of group theory, a central concept is the normal subgroup. Imagine a set within a larger mathematical structure, which is a group, behaving in a unique way when combined with elements from the entire group. A subgroup, denoted as \( H \), is considered normal if for every member \( g \) in the larger group \( G \), the subgroup remains unchanged when conjugated by \( g \). Mathematically, this is represented as \( gHg^{-1} = H \), which leads to the notation \( H \trianglelefteq G \).
Normal subgroups are vital because they allow the creation of new structures known as quotient groups. These subgroups act as building blocks, making it possible to divide the group \( G \) by \( H \) in a systematic way. This division maintains the essential properties of the group \( G \) but removes complexities introduced by the elements of \( H \).

• Normal Subgroup Notation: \( H \trianglelefteq G \).
• Conjugation property: \( gHg^{-1} = H \).
• Forms quotient groups: \( G/H \).

Abelian Group

An abelian group is a group that follows the commutative property in its operations. This means that within an abelian group \( G \), for any two elements \( x \) and \( y \), the result of their operation is the same regardless of the order. Simply put, \( xy = yx \). When discussing group operations, this property greatly simplifies the interactions of elements within \( G \).
Abelian groups are ubiquitous in many areas of mathematics and science due to their straightforward, predictable behavior. They mirror the basic properties of addition in arithmetic, where the order of numbers does not affect the result.
Key points about abelian groups include:

• Commutativity: \( xy = yx \) for all \( x, y \in G \).
• Simplified calculations and solutions in mathematical problems.
• Represents many real-world phenomena due to its simple nature.

Cosets

The concept of a coset emerges when we partition a group \( G \) by a subgroup \( H \). Cosets are essentially the pieces you get when dividing a group. Specifically, for a group \( G \) and a subgroup \( H \), a left coset of \( H \) in \( G \) is the set formed by multiplying each element of \( H \) by a fixed element \( g \) in \( G \). This is denoted as \( gH \). Similarly, there is a right coset denoted as \( Hg \).
Cosets are integral to understanding how groups are structured, especially when discussing quotient groups like \( G/H \). They form the distinct "pieces" or "classes" of the quotient group, where each coset is represented by an element from the group \( G \).

• Left Coset: \( gH \).
• Right Coset: \( Hg \).
• Essential for quotient groups \( G/H \).

Group Theory

Group theory is a broad mathematical framework that studies groups, structures consisting of a set equipped with an operation that combines any two elements to form another element within the same group. This area of mathematics explores how these elements relate and work together.
Central to group theory is the idea of identity, inverse, and operation. Every group must have an identity element, each element must have an inverse, and the operation (like addition or multiplication) must be associative for it to be considered a group.
Group theory provides a language and set of tools to explore symmetries in geometry, solutions to equations, and even aspects of quantum physics. By understanding the principles of group theory, mathematicians can abstractly tackle and solve complex problems in diverse fields.

• Key elements: Identity, inverse, operation.
• Underlying structure of many mathematical and physical concepts.
• Enables abstraction and solution of complex problems.