(a) Let $H$ be the subgroup of $S_5$ consisting of those permutations that fix the letters $2$ and $4$. For each of the following pairs of elements of $S_5$, determine whether or not they belong to the same left $H$-coset.

$$(i) \quad \quad f_1 = \displaystyle\left(\begin{array}{ccccc} 1 &2 &3 &4 &5 \cr 5 &2 &3 &4 &1 \cr \end{array}\right), \quad g_1 = \displaystyle\left(\begin{array}{ccccc} 1 &2 &3 &4 &5 \cr 1 &4 &5 &3 &2 \cr \end{array}\right)$$


$$(ii) \quad \quad f_2 = \displaystyle\left(\begin{array}{ccccc} 1 &2 &3 &4 &5 \cr 2 &1 &3 &4 &5 \cr \end{array}\right), \quad g_2 = \displaystyle\left(\begin{array}{ccccc} 1 &2 &3 &4 &5 \cr 5 &1 &3 &4 &2 \cr \end{array}\right)$$


(b) Let $K$ be the subgroup of $S_5$ generated by $( 1 2 3 4 )$ and $( 1 2 )( 3 4 )$. For each of the following pairs of elements of $S_5$, determine whether or not they belong to the same left $K$-coset.

$$(i) \quad \quad f_3 = \displaystyle\left(\begin{array}{ccccc} 1 &2 &3 &4 &5 \cr 1 &2 &4 &5 &3 \cr \end{array}\right), \quad g_3 = \displaystyle\left(\begin{array}{ccccc} 1 &2 &3 &4 &5 \cr 4 &5 &1 &3 &2 \cr \end{array}\right)$$


$$(ii) \quad \quad f_4 = \displaystyle\left(\begin{array}{ccccc} 1 &2 &3 &4 &5 \cr 1 &5 &2 &3 &4 \cr \end{array}\right), \quad g_4 = \displaystyle\left(\begin{array}{ccccc} 1 &2 &3 &4 &5 \cr 3 &1 &5 &2 &4 \cr \end{array}\right)$$