Problem 12 - Algebra

Question

Let \(G\) be the group of upper triangular matrices with non-zero diagonal elements. Let \(H\) be the subgroup consisting of those matrices whose diagonal element is 1 . (Actually prove that \(H\) is a subgroup). How would you describe the factor group \(G / H ?\)

Step-by-Step Solution

Answer

The subgroup H consists of upper triangular matrices with diagonal elements equal to 1 and can be represented as: \[H = \left\{ \begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix} \mid b \in \mathbb{R} \right\}\] H is indeed a subgroup of G, as it satisfies the criteria for identity element, inverses, and closure. The factor group G/H consists of cosets of H in G and can be described as pairs of non-zero real numbers (a, c), representing the diagonal elements of the upper triangular matrices in G: \[G/H \cong \mathbb{R}^* \times \mathbb{R}^*\]

Step 1: Define the group G and subgroup H

The group G consists of upper triangular matrices with non-zero diagonal elements. An upper triangular matrix looks like this: \[G = \left\{ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \mid a,b,c \in \mathbb{R}, a,c \neq 0 \right\}\] The subgroup H consists of those matrices whose diagonal elements are 1: \[H = \left\{ \begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix} \mid b \in \mathbb{R} \right\}\]

Step 2: Prove that H is a subgroup

To show that H is a subgroup of G, we need to check the following criteria: 1. Identity element: There exists an identity element e in H such that for any element h in H, e*h = h*e = h; 2. Inverses: For any element h in H, there exists an inverse h^(-1) also in H such that h*h^(-1) = h^(-1)*h = e; 3. Closure: For any two elements h1 and h2 in H, the product h1 * h2 is also in H. Let's check these one by one: 1. Identity element: The identity element for matrix multiplication is the identity matrix: \[\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\] This matrix is in H since its diagonal elements are 1, so the identity element exists in H. 2. Inverses: Let's consider an element h in H: \[h = \begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix}\] To find its inverse, we want to find a matrix such that their product is an identity matrix: \[h^{-1} = \begin{pmatrix} 1 & -b \\ 0 & 1 \end{pmatrix}\] Multiplying h and h^(-1): \[h * h^{-1} = \begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix} * \begin{pmatrix} 1 & -b \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\] Hence, h^(-1) is also in H, and this holds for any element in H. 3. Closure: Let h1 and h2 be any two elements in H: \[h1 = \begin{pmatrix} 1 & b1 \\ 0 & 1 \end{pmatrix}, h2 = \begin{pmatrix} 1 & b2 \\ 0 & 1 \end{pmatrix}\] Now let's compute the product of h1 and h2: \[h1 * h2 = \begin{pmatrix} 1 & b1 \\ 0 & 1 \end{pmatrix} * \begin{pmatrix} 1 & b2 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & b1+b2 \\ 0 & 1 \end{pmatrix}\] Since b1 and b2 are real numbers, their sum is also a real number, so (b1+b2) ∈ ℝ, ensuring that the resulting matrix is in H. Hence, H is a subgroup of G since it meets all the criteria.

Step 3: Describe the factor group G/H

The factor group G/H consists of the cosets of H in G. Let's consider a matrix g in G: \[g = \begin{pmatrix} a & b \\ 0 & c \end{pmatrix}\] Then the coset gH contains all elements obtained by multiplying g by elements of H: \[gH = \left\{ g * h \mid h \in H \right\} = \left\{ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} * \begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix} \mid x \in \mathbb{R} \right\} = \left\{ \begin{pmatrix} a & b+ax \\ 0 & c \end{pmatrix} \mid x \in \mathbb{R} \right\}\] The factor group G/H consists of these cosets: \[G/H = \left\{ gH \mid g \in G \right\}\] Since the coset space is defined by the values of a and c, we can describe the factor group G/H as a set of pairs of non-zero real numbers (a, c): \[G/H \cong \mathbb{R}^* \times \mathbb{R}^*\] Here, \(\mathbb{R}^*\) denotes the non-zero real numbers. In conclusion, the factor group G/H can be described as pairs of non-zero real numbers, representing the diagonal elements of the upper triangular matrices in G.

Key Concepts

Upper Triangular Matrix

An upper triangular matrix is a type of square matrix where all the elements below the main diagonal are zero. This means the matrix has a form where non-zero elements only appear on or above the diagonal. For example:

• Given a 2x2 upper triangular matrix: \[ \begin{pmatrix} a & b \ 0 & c \end{pmatrix} \]

In this matrix, the position where we typically expect (2,1) to have a value is zero. Only the diagonal elements and those above it, like \(a, b,\) and \(c\), can be non-zero.
This structure simplifies many calculations, such as matrix multiplication and finding determinants. Determinants, for example, can be found simply by multiplying the diagonal elements in an upper triangular matrix. This simplicity and structured constraints make upper triangular matrices valuable in various mathematical applications.

Subgroup

A subgroup is essentially a group within a group. It shares the properties of a group and fits completely within a larger group, maintaining the operations and identity of the overall larger group. To be a subgroup, a subset must fulfill certain conditions:

• Identity element: The subgroup must contain the group's identity element.
• Closure: The operation defined for the group still applies within the subgroup. If you take two elements from the subgroup and perform a group operation (like multiplication), the result must also be in the subgroup.
• Inverses: Every element in the subgroup must have an inverse also within the subgroup.

In our example, the subgroup \(H\) contains upper triangular matrices with ones on the diagonal. This fulfils the idea of a subgroup since any matrix in \(H\) can combine with others within \(H\) using matrix multiplication, keeping all operations enclosed within \(H\).
This containment allows us to form new concepts within the larger group such as quotients or factor groups.

Cosets

Cosets are key players when we study the structure of groups, especially in forming factor groups. In simple terms, a coset is formed by taking an element from a group and multiplying it with all elements of a subgroup. This leads to two types of cosets if we use a group \(G\) and a subgroup \(H\):

• Left coset: Generated by a fixed element \(g\) in \(G\) as \(gH\), which means multiply \(g\) with every element in \(H\).
• Right coset: Created similarly but as \(Hg\), meaning we compute every element in \(H\) multiplied by \(g\).

In our exercise, by talking about \(G/H\), we focus on left cosets. Every element in \(G\) can generate a unique coset that fits into this factorization. Notably, distinct cosets are either non-overlapping or entirely the same. This helps classify group elements into distinct, manageable sections.

Real Numbers

Real numbers are the broad class of numbers used in mathematics that can fill the entire number line. They include all the rational numbers (such as integers like -1, 0, 1, fractions like 1/2) and the irrational numbers (numbers that cannot be represented as simple fractions, like \(\pi\) and \(\sqrt{2}\)).

• Non-zero real numbers: These are real numbers excluding zero. This subset is crucial when we describe the factor group \(G/H\) because diagonal elements in upper triangular matrices must be non-zero to ensure invertibility.

Real numbers are fundamental in group theory to describe elements, verify identities, and perform calculations. In our context of upper triangular matrices, elements like \(a, b,\) and \(c\) are taken from the real numbers, each having implications on group properties and behaviors, such as ensuring the diagonal elements don't collapse into singular or undefined forms.