Question 4 - A-Level Maths

Given that $e$ is the identity element and that $$p ^ { * } p = s \quad s ^ { * } s = r \quad p ^ { * } p ^ { * } p = q$$

show that
1. $p ^ { * } q = r$
2. $s ^ { * } p = q$
Hence complete the Cayley table below.
* $e$ $\boldsymbol { p }$ $\boldsymbol { q }$ $r$ $s$
$e$
$\boldsymbol { p }$
$\boldsymbol { q }$
$\boldsymbol { r }$
$S$
A spare table can be found on page 11 if you need to rewrite your Cayley table.
Use your table to find $p ^ { * } q ^ { * } r ^ { * } s$ A student states that there is a subgroup of $A$ of order 3
Comment on the validity of this statement, giving a reason for your answer. \includegraphics[max width=\textwidth, alt={}, center]{989d779e-c40a-4658-ad98-17a37ab1d9e1-11_2464_74_304_36}
Only use this grid if you need to rewrite the Cayley table.
* $e$ $\boldsymbol { p }$ $\boldsymbol { q }$ $r$ $s$
$e$
$\boldsymbol { p }$
$\boldsymbol { q }$
$\boldsymbol { r }$
$S$