Mathematics A, Further H235: Groups Topical Past Paper Questions

The set S consists of all $ 2 \times 2 $ matrices having determinant 1 or -1. For instance, the matrices $ \mathbf{P} = \begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix} $, $ \mathbf{Q} = -\begin{pmatrix} \frac{1}{2} & \frac{\sqrt…

The group G, of order 12, consists of the set $ \{1, 2, 4, 5, 8, 10, 13, 16, 17, 19, 20, x\} $ under the operation of multiplication modulo 21. The identity of G is the element 1. The element x is an integer, $ 0 An incomplete copy of the Cayley table for $G$ is shown below: (a)…

The set S consists of the following four complex numbers. $$ \sqrt{3}+\mathrm{i}\quad-\sqrt{3}-\mathrm{i}\quad1-\mathrm{i}\sqrt{3}\quad-1+\mathrm{i}\sqrt{3} $$ For $ z_1, z_2 \in S $, the binary operation $ \circ $ is defined by $ z_1 \circ z_2 = \frac{1}{4}(1 + i\sqrt{3})z_1z_2…