Let's address the problem step by step.

Problem 2: Finding the order of $G$

Given:

• $\mathbb{F}_7 = \mathbb{Z}/7\mathbb{Z}$ is the finite field with 7 elements.
• $H = \langle 2 \rangle = \{ 2^n \mid n \in \mathbb{Z} \}$ is the multiplicative subgroup of $\mathbb{F}_7^*$ generated by 2.
• $G = \left\{ \left( \begin{matrix} a & b \\ 0 & a \end{matrix} \right) \mid a \in H, \, b \in \mathbb{F}_7 \right\}$.

Step 1: Determine the structure of $H$

• $H = \langle 2 \rangle$ is the group generated by 2 under multiplication modulo 7.
• The elements of $\mathbb{F}_7^*$ are $\{1, 2, 3, 4, 5, 6\}$.
• $2^1 \equiv 2$, $2^2 \equiv 4$, $2^3 \equiv 8 \equiv 1 \mod 7$.

Thus, $H = \{1, 2, 4\}$, which has order 3.

Step 2: Count the number of elements in $G$

• For each $a \in H$ (3 choices), $b$ can be any element of $\mathbb{F}_7$ (7 choices).

Hence, the order of $G$ is: $|G| = |H| \times |\mathbb{F}_7| = 3 \times 7 = 21$

Problem 3: Show that the map $\phi: \mathbb{F}_7 \to G$ given by $\phi(b) = \left( \begin{matrix} 1 & b \\ 0 & 1 \end{matrix} \right)$ is a group homomorphism

Step 1: Verify the group operation

• $\mathbb{F}_7$ under addition.
• $G$ under matrix multiplication.

Step 2: Verify the homomorphism property

For $b_1, b_2 \in \mathbb{F}_7$, we need to show: $\phi(b_1 + b_2) = \phi(b_1) \cdot \phi(b_2)$ where: $\phi(b_1 + b_2) = \left( \begin{matrix} 1 & b_1 + b_2 \\ 0 & 1 \end{matrix} \right)$ and: $\phi(b_1) \cdot \phi(b_2) = \left( \begin{matrix} 1 & b_1 \\ 0 & 1 \end{matrix} \right) \cdot \left( \begin{matrix} 1 & b_2 \\ 0 & 1 \end{matrix} \right) = \left( \begin{matrix} 1 \times 1 + b_1 \times 0 & 1 \times b_2 + b_1 \times 1 \\ 0 \times 1 + 1 \times 0 & 0 \times b_2 + 1 \times 1 \end{matrix} \right) = \left( \begin{matrix} 1 & b_1 + b_2 \\ 0 & 1 \end{matrix} \right)$

Since $\phi(b_1 + b_2) = \phi(b_1) \cdot \phi(b_2)$, the map $\phi$ is indeed a group homomorphism.

Summary

1. The order of $G$ is 21.
2. The map $\phi: \mathbb{F}_7 \to G$ given by $\phi(b) = \left( \begin{matrix} 1 & b \\ 0 & 1 \end{matrix} \right)$ is a group homomorphism.

Would you like more details on any of these steps, or do you have any questions?

Relative Questions

1. How would the problem change if $\mathbb{F}_7$ were replaced by $\mathbb{F}_5$?
2. Can you describe the kernel of the homomorphism $\phi$?
3. How would you describe the image of the homomorphism $\phi$?
4. What is the significance of the group $G$ in the context of linear algebra?
5. How does the order of $H$ influence the structure of $G$?
6. What are some examples of other matrices that belong to $G$?
7. How would the analysis change if $H$ had more or fewer elements?
8. Can you identify other homomorphisms from $\mathbb{F}_7$ to $G$?

Tip

When checking if a map is a homomorphism, always ensure to verify the group operation compatibility by explicitly performing the operations on both sides.