Question 4 - A-Level Maths

Exam Board	OCR MEI
Module	Further Extra Pure (Further Extra Pure)
Year	2020
Session	November
Topic	Groups

In each of the following cases, a set $G$ and a binary operation ∘ are given. The operation ∘ may be assumed to be associative on $G$. Determine which, if any, of the other three group axioms are satisfied by ( $G , \circ$ ) and which, if any, are not satisfied.
1. $G = \{ 2 n + 1 : n \in \mathbb { Z } \}$ and $\circ$ is addition.
2. $G = \{ a + b \sqrt { 2 } : a , b \in \mathbb { Z } \}$ and ∘ is multiplication.
3. $G$ is the set of all real numbers and ∘ is multiplication.
A group $M$ consists of eight $2 \times 2$ matrices under the operation of matrix multiplication. Five of the eight elements of $M$ are as follows. $$\frac { 1 } { \sqrt { 2 } } \left( \begin{array} { l l } 1 & \mathrm { i }
\mathrm { i } & 1 \end{array} \right) \quad \frac { 1 } { \sqrt { 2 } } \left( \begin{array} { r r } - 1 & \mathrm { i }
\mathrm { i } & - 1 \end{array} \right) \quad \frac { 1 } { \sqrt { 2 } } \left( \begin{array} { r r } 1 & - \mathrm { i }
- \mathrm { i } & 1 \end{array} \right) \quad \left( \begin{array} { l l } 0 & \mathrm { i }
\mathrm { i } & 0 \end{array} \right) \quad \left( \begin{array} { l l } 1 & 0
0 & 1 \end{array} \right)$$
1. Find the other three elements of $M$.
  $( N , * )$ is another group of order 8, with identity element $e$. You are given that $N = \langle a , b , c \rangle$ where $a * a = b * b = c * c = e$.
2. State whether $M$ and $N$ are isomorphic to each other, giving a reason for your answer.