The elements of the set \(P = \{ 1,3,9,11 \}\) are combined under the binary operation, *, defined as multiplication modulo 16.
Demonstrate associativity for the elements \(3,9,11\) in that order.
Assuming associativity holds in general, show that \(P\) forms a group under the binary operation *.
Write down the order of each element.
Write down all subgroups of \(P\).
Show that the group in part (i) is cyclic.
Now consider a group of order 4 containing the identity element \(e\) and the two distinct elements, \(a\) and \(b\), where \(a ^ { 2 } = b ^ { 2 } = e\). Construct the composition table. Show that the group is non-cyclic.
Now consider the four matrices \(\mathbf { I } , \mathbf { X } , \mathbf { Y }\) and \(\mathbf { Z }\) where
$$\mathbf { I } = \left( \begin{array} { l l }
1 & 0
0 & 1
\end{array} \right) , \mathbf { X } = \left( \begin{array} { r r }
1 & 0
0 & - 1
\end{array} \right) , \mathbf { Y } = \left( \begin{array} { r r }
- 1 & 0
0 & 1
\end{array} \right) , \mathbf { Z } = \left( \begin{array} { r r }
- 1 & 0
0 & - 1
\end{array} \right) .$$
The group G consists of the set \(\{ \mathbf { I } , \mathbf { X } , \mathbf { Y } , \mathbf { Z } \}\) with binary operation matrix multiplication. Determine which of the groups in parts (a) and (b) is isomorphic to G, and specify the isomorphism.
The distinct elements \(\{ p , q , r , s \}\) are combined under the binary operation \({ } ^ { \circ }\). You are given that \(p ^ { \circ } q = r\) and \(q ^ { \circ } p = s\).
By reference to the group axioms, prove that \(\{ p , q , r , s \}\) is not a group under \({ } ^ { \circ }\).
Option 5: Markov chains
\section*{This question requires the use of a calculator with the ability to handle matrices.}