Question

Let \( F = \{\omega \in \mathbb{C} : \omega^{2020} = 1\}. \) Consider the groups \\ \[ G = \left\{ \begin{pmatrix} \omega & z \\ 0 & 1 \end{pmatrix} : \omega \in F, z \in \mathbb{C} \right\} \text{and} H = \left\{ \begin{pmatrix} 1 & z \\ 0 & 1 \end{pmatrix} : z \in \mathbb{C} \right\} \] under matrix multiplication. Then the number of cosets of \( H \) in \( G \) is \\

• A. 1010
• B. 2019
• C. 2020
• D. infinite

Hint:

For groups of upper-triangular matrices, distinct diagonal entries usually define distinct cosets, especially when the subgroup fixes the diagonal.

Answer 1

The Correct Option is C

Step 1: Understand the structure of the groups.
Each element of \( G \) is of the form \[ \begin{pmatrix} \omega & z \\ 0 & 1 \end{pmatrix}, \text{where } \omega \in F = \{\text{2020th roots of unity}\}, \; z \in \mathbb{C}. \] Hence, \( G \) consists of all such upper-triangular matrices with unit determinant in the lower diagonal entry.
Each element of \( H \) is of the form \[ \begin{pmatrix} 1 & z \\ 0 & 1 \end{pmatrix}, \text{where } z \in \mathbb{C}. \] Step 2: Group multiplication rule.
For two elements of \( G \): \[ \begin{pmatrix} \omega_1 & z_1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} \omega_2 & z_2 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} \omega_1\omega_2 & \omega_1z_2 + z_1 \\ 0 & 1 \end{pmatrix}. \] Thus, \( G \) forms a group under matrix multiplication. Step 3: Identify \( H \) as a subgroup of \( G. \)
All elements of \( H \) correspond to the case \( \omega = 1 \). Hence, \( H \) is indeed a subgroup of \( G \) containing all matrices with \( \omega = 1 \).
Step 4: Find the left cosets of \( H \) in \( G. \) \\ Consider \( g = \begin{pmatrix} \omega & 0 \\ 0 & 1 \end{pmatrix} \in G. \) Then a left coset is: \[ gH = \begin{pmatrix} \omega & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & z \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} \omega & \omega z \\ 0 & 1 \end{pmatrix}. \] If \( \omega_1 \neq \omega_2, \) then \( g_1H \neq g_2H. \) Thus, each distinct \( \omega \in F \) gives a distinct coset. Step 5: Count distinct cosets.
Since \( \omega \) takes 2020 distinct values (the 2020th roots of unity), the number of distinct cosets of \( H \) in \( G \) is equal to \( |F| = 2020. \)

Final Answer: \[ \boxed{2020} \]