Left cosets of a non-normal subgroup of $A_4$ do not form a group

Let \(H=\{(1),(12)(34)\}\) in \(A_4\).

1. Show that \(H\) is not normal in \(A_4\).

2. Referring to the multiplication table for \(A_4\) in Table 5.1 on page 111, show that, although \(\alpha_6H=\alpha_7H\) and \(\alpha_9H=\alpha_{11}H\), it is not true that \(\alpha_6\alpha_9H=\alpha_7\alpha_{11}H\). Explain why this proves that the left cosets of \(H\) do not form a group under coset multiplication.

In Table 5.1, \(\alpha_6 = (243)\), \(\alpha_7 = (142)\), \(\alpha_9 = (132)\) and \(\alpha_{11} = (234)\), as identified in the source’s solution.