Hw8 Problem 1

(a):

Let $H = \{(1), (12)(34), (13)(24), (14)(23) \}$. Find the left cosets of $H$ in $A_{4}$.

$H = {(1),(12)(34),(13)(24),(14)(23)}$

$A_{4} = {(1),(12)(34),(13)(24),(14)(23),(123),(124),(132),(134),(142),(143),(234),(243)}$

(1)
\begin{align} (1) \cdot H = \{ (1), (12)(34), (13)(24), (14)(23) \} = H \end{align}
(2)
\begin{align} (12)(34) \cdot H = \{(12)(34), (1), (14)(23), (13)(24) \} = H \end{align}
(3)
\begin{align} (13)(24) \cdot H = \{(13)(24), (14)(23), (1), (12)(34) \} = H \end{align}
(4)
\begin{align} (14)(23) \cdot H = \{(14)(23), (13)(24), (12)(34), (1) \} = H \end{align}
(5)
\begin{align} (123) \cdot H = \{(123), (134), (243), (142) \} \end{align}
(6)
\begin{align} (124) \cdot H = \{(124), (143), (132), (234) \} \end{align}
(7)
\begin{align} (132) \cdot H = \{(132), (234), (124), (143) \} \end{align}
(8)
\begin{align} (134) \cdot H = \{(134), (123), (143), (243) \} \end{align}
(9)
\begin{align} (142) \cdot H = \{(142), (243), (134), (123) \} \end{align}
(10)
\begin{align} (143) \cdot H = \{(143), (124), (234), (132) \} \end{align}
(11)
\begin{align} (234) \cdot H = \{(234), (132), (143), (124) \} \end{align}
(12)
\begin{align} (243) \cdot H = \{(243), (142), (123), (134) \} \end{align}

Left cosets = $H, \{ (123),(134),(142),(243) \} , \{ (124),(132),(143),(234) \}$

(b):

The index of $H$ in $G$, where $H$ is a subgroup of $G$,

(13)
\begin{align} = \frac{|G|}{|H|} = \frac{|S^{4}|}{|H|} = \frac{24}{4} = 6 \end{align}