Question 35
Let $G$ be the set of matrices
$$
\begin{pmatrix} a & b \\ \hat{0} & \hat{1} \end{pmatrix}, \quad a, b \in \mathbb{Z}_7, a \neq \hat{0}.
$$
a) Show that $G$ is a group with respect to the matrix multiplication.
b) Prove that there exists no proper homomorphism from $G$ to $\mathbb{Z}_7$.
$$
\begin{pmatrix} a & b \\ \hat{0} & \hat{1} \end{pmatrix}, \quad a, b \in \mathbb{Z}_7, a \neq \hat{0}.
$$
a) Show that $G$ is a group with respect to the matrix multiplication.
b) Prove that there exists no proper homomorphism from $G$ to $\mathbb{Z}_7$.