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), \ Quadad \ sigma_ {2} = \ left (\ begin {array} {cc} 0 & -i \\ i & 0 \ End {Array} \ Right), \ Quad \ Sigma_ {3} = \ Left (\ Begin {Array} {cc} 1 & 0 & -1 \ End {Array} \ Right) \ Quad \) are called Pauli matrices. Denine \ (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 \} \) the group regarding the operation of the multiplication of matrices. 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 the normal subgroups in \ (q_ {8} \).