Question

Find the group \(\mathcal{G}\) generated under matrix multiplication by the matrices $$ \mathrm{A}=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right), \quad \mathrm{B}=\left(\begin{array}{ll} 0 & i \\ i & 0 \end{array}\right) $$ Determine its proper subgroups, and verify for each of them that its cosets exhaust \(\mathcal{G}\).

Short answer

The group \( \mathcal{G} \) has elements \( I, A, B, -I \). Its proper subgroups are \( H_1 \) and \( H_2 \), with cosets \( \mathcal{G}/H_1 = \ { I, A, B, -I \} \) and \( \mathcal{G}/H_2 = \{ H_2, A H_2, B H_2 \} \).

Step-by-step solution

- Understand Basis Matrices

Examine the given matrices. Matrix \( A = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} \) and Matrix \( B = \begin{pmatrix} 0 & i \ i & 0 \end{pmatrix} \), where \( i \) is the imaginary unit.

- Check Properties of Basis Matrices

Check if the given matrices square to identity: \[ A^2 = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}, \quad B^2 = \begin{pmatrix} 0 & i \ i & 0 \end{pmatrix} \begin{pmatrix} 0 & i \ i & 0 \end{pmatrix} = \begin{pmatrix} -1 & 0 \ 0 & -1 \end{pmatrix} \]

- Determine Group Elements

Calculate all possible products of the matrices including identity. Possible matrices are: Identity \( I=\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}, A, B, AB, BA, B^2, A^2 = I, B^2 = -I \)

- Identify Distinct Elements

Identify unique matrices: \[ \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}, \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & i \ i & 0 \end{pmatrix}, \begin{pmatrix} -1 & 0 \ 0 & -1 \end{pmatrix} \]

- Find Subgroups

List subgroups: Proper subgroups are: \[ H_1 = \left\{ \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \right\}, \, H_2 = \left\{ \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}, \begin{pmatrix} -1 & 0 \ 0 & -1 \end{pmatrix} \right\} \]

- Verify Cosets

Check cosets: 1. For \( H_1 \), cosets are \( \mathcal{G}/H_1 = \{ I, A, B, -I \} \)2. For \( H_2 \), cosets are \( \mathcal{G}/H_2 = \{ H_2, \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} H_2, \begin{pmatrix} 0 & i \ i & 0 \end{pmatrix} H_2 \} \)

Key concepts

matrix multiplication

Matrix multiplication is a key operation in linear algebra. It involves combining two matrices to produce a third matrix. To multiply two matrices, you perform the dot product of corresponding rows and columns. Given matrices \A\ and \B\, the element in the \((i,j)\)-th position of the product matrix is calculated as: \[ (AB)_{ij} = \sum_k A_{ik} B_{kj} \] This technique is essential when working with transformations, rotations, and other applications in math and science. It helps us understand the structure and properties of groups generated by matrices, like the one described in the original exercise.

group theory

Group theory is a mathematical field that studies algebraic structures known as groups. A group comprises a set equipped with an operation (like matrix multiplication) that combines any two elements to form a third element. Groups must satisfy four fundamental properties:

• Closure: Combining two elements of the group results in another group element.
• Associativity: The way elements are grouped in an operation does not affect the result.
• Identity: There is an element (the identity) that, when combined with any element, leaves it unchanged.
• Inverses: Each element has an inverse that, when combined with it, results in the identity element.

Understanding these helps grasp the concept of subgroups and cosets as used in the original exercise.

proper subgroups

A proper subgroup is any subgroup of a group that is not equal to the whole group but is still a subset. For instance, if \mathcal{G}\ is the group generated by matrices \A\ and \B\, its proper subgroups can be smaller sets containing elements of \mathcal{G}\ but not all elements.
In the exercise, \(H_1\) and \(H_2\) are proper subgroups of \mathcal{G}\, where \[H_1 = \left\{ \begin{pmatrix} 1 \ 0 \ 0 \ 1 \end{pmatrix} \right\} \] \(H_2\) contains the identity and another matrix. Proper subgroups are foundational to understanding the structure and hierarchy within the group.

cosets

Cosets are sets formed by multiplying all elements of a subgroup by a fixed element from the group. For a subgroup \H\ in \G\, and an element \g\ in \G\, the left coset of \H\ with respect to \g\ is: \[ gH = \{gh : h \in H \} \] Cosets partition the group into equal-sized, non-overlapping sets. In the provided exercise, we verified cosets for proper subgroups \H_1\ and \H_2\. Understanding cosets helps in exploring symmetrical structures and classifications within the group.

identity matrix

The identity matrix is a special type of matrix that, when multiplied by another matrix, does not change that matrix. For an \(n \times n\) matrix, the identity matrix \I\ has 1's on the diagonal and 0's elsewhere: \[ I = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \] This matrix plays a crucial role in matrix groups as the identity element, satisfying the 'identity' property of group theory. In the exercise, it appeared as it ensured the group's structure was maintained through matrix multiplications.