- The group \(\mathrm { S } _ { 4 }\) is the set of all possible permutations that can be performed on the four numbers 1, 2, 3 and 4, under the operation of composition.
For the group \(\mathrm { S } _ { 4 }\)
- write down the identity element,
- write down the inverse of the element \(a\), where
$$a = \left( \begin{array} { l l l l }
1 & 2 & 3 & 4
3 & 4 & 2 & 1
\end{array} \right)$$ - demonstrate that the operation of composition is associative using the following elements
$$a = \left( \begin{array} { l l l l }
1 & 2 & 3 & 4
3 & 4 & 2 & 1
\end{array} \right) \quad b = \left( \begin{array} { l l l l }
1 & 2 & 3 & 4
2 & 4 & 3 & 1
\end{array} \right) \quad \text { and } c = \left( \begin{array} { l l l l }
1 & 2 & 3 & 4
4 & 1 & 2 & 3
\end{array} \right)$$ - Explain why it is possible for the group \(\mathrm { S } _ { 4 }\) to have a subgroup of order 4 You do not need to find such a subgroup.