Problem 4 - A Book of Abstract Algebra

Question

For any pair of real numbers \(a \neq 0\) and \(b\), define a function \(f_{a, b}\) as follows: $$ f_{a, b}(x)=a x+b $$ Let \(G=\left\\{f_{a, b}: a \in \mathbb{R}, b \in \mathbb{R}, a \neq 0\right\\} .\) Show that \(G\) is a subgroup of \(S_{R}\)

Step-by-Step Solution

Answer

G is a subgroup of \( S_R \) because it is closed under composition, contains the identity \( f_{1,0} \), and every element has an inverse in \( G \).

Step 1: Define the Group

The set of functions \( G \) consists of linear functions of the form \( f_{a, b}(x) = ax + b \) where \( a eq 0 \). We need to show that \( G \) satisfies the properties of a subgroup under function composition in the symmetric group \( S_R \) of all bijections from \( \mathbb{R} \) to itself.

Step 2: Verify Closure

To show closure, we need to confirm that the composition of any two functions in \( G \) results in another function in \( G \). Take two functions \( f_{a_1, b_1}(x) = a_1x + b_1 \) and \( f_{a_2, b_2}(x) = a_2x + b_2 \). Their composition is: \((f_{a_1, b_1} \circ f_{a_2, b_2})(x) = f_{a_1, b_1}(a_2x + b_2) = a_1(a_2x + b_2) + b_1 = a_1a_2x + a_1b_2 + b_1\). This is of the form \(f_{a_1a_2, a_1b_2 + b_1} \), so closure is satisfied.

Step 3: Check for Identity Element

The identity function in the set of functions is \( f_{1, 0}(x) = x \). Substituting for the function in \( G \), we get \( f_{1, 0}(x) = 1 \cdot x + 0 = x \) which leaves any element unchanged under composition, fulfilling the identity condition.

Step 4: Establish Existence of Inverses

For each function \( f_{a, b}(x) = ax + b \), we need to find an inverse function \( g(x) \) such that \( f_{a, b}(g(x)) = g(f_{a, b}(x)) = x \). Solving \( ax + b = y \), we find \( x = \frac{y - b}{a} \). Hence, the inverse is \( g(x) = \frac{x - b}{a} \), which is of the form \( f_{\frac{1}{a}, -\frac{b}{a}} \). Thus, every function has an inverse within \( G \).

Step 5: Conclude Subgroup Properties

Since \( G \) satisfies closure, contains the identity element, and every element has an inverse also in \( G \), \( G \) is a subgroup of \( S_R \).

Key Concepts

Subgroup

In Abstract Algebra, the concept of a subgroup is very important. A subgroup is a subset of a group that itself satisfies the properties that define a group. These properties include:

• Closure: If you combine two elements in the subgroup, their result should also be in the subgroup.
• Identity: There should be an element in the subgroup that doesn’t change any other element when combined with it.
• Inverses: For every element in the subgroup, there should be another element in the subgroup that combines with it to yield the identity element.

To determine if a set of elements is a subgroup, you check these three properties. In our exercise, we have a set of functions that meet these conditions under function composition, making it a valid subgroup of the larger group of all bijections.

Function Composition

Function composition is a crucial operation in mathematics, especially in understanding how functions interact with each other. It is essentially applying one function and then applying another function to the result. For example, if you have a function \(f(x)\) and another function \(g(x)\), the composition \((f \circ g)(x)\) means you first apply \(g(x)\) and then \(f\) to that result.

• This operation is associative, meaning the order in which you compose multiple functions doesn't matter.
• The result of composing two functions is another function.
• Function composition allows exploring complex mappings by building them from simpler ones.

In our exercise, composing the functions \(f_{a, b}(x) = ax + b\) results in another function of the same form, demonstrating the property of closure, which is essential to be a subgroup under composition.

Linear Functions

Linear functions form the foundation of many mathematical concepts. They are functions that graph as straight lines. The general form of a linear function is \(f(x) = ax + b\), where \(a\) and \(b\) are constants, and \(a eq 0\).

• Constant \(a\): This determines the slope of the line; it dictates how steep the line is.
• Constant \(b\): This represents the y-intercept of the line, which is where the line crosses the y-axis.

Linear functions are translatable via shifts and stretches based on these constants. In our study of subgroups, the set of functions \(f_{a, b}(x) = ax + b\) ensures that any combination of these shifts still results in a linear function, an essential characteristic for closure in the subgroup.

Symmetric Group

The symmetric group, often denoted as \(S_n\), is a key concept in Abstract Algebra. It is the group consisting of all possible permutations of a set of \(n\) elements. The symmetric group \(S_R\) in our exercise refers to the group of all bijections from the real numbers to themselves.

• Bijections: A function that is both injective (one-to-one) and surjective (onto), which means it can be reversed uniquely.
• Permutations: Different arrangements or orderings of elements in a set.

In the context of our exercise, the set \(G\) composed of linear functions is considered a subgroup because it fulfills all subgroup requirements when viewed as a special kind of permutation (or bijection) on the real numbers. Symmetric groups help understand the structure and relationship between these functions.