1.6.83 Fields, Groups, Rings

condition: Complex matrices of size \( 2 \times 2 \) \( \sigma_{1}=\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right), \quad \sigma_{2}=\left(\begin{array}{cc}0 & -i \\ i & 0\end{array}\right), \quad \sigma_{3}=\left(\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right) \quad \) are called Pauli matrices. Let us denote \( E=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right) \). a) Prove that the set \( Q_{8}=\left\{ \pm E, \pm i \sigma_{1}, \pm i \sigma_{2}, \pm i \sigma_{3}\right\} \) is a group under the operation of matrix multiplication. b) Find the order of all elements of the group \(Q_{8}\). Show that \( Q_{8} \) is not a cyclic group. c) Find all subgroups in \(Q_{8}\). d) Find all normal subgroups in \( Q_{8} \).