Give an example of subgroups \[ H\trianglelefteq K\trianglelefteq G \] with \(H\) not normal in \(G\).
Solution
Take \[ G=A_4, \qquad K=V_4=\{e,(12)(34),(13)(24),(14)(23)\}, \] and \[ H=\langle(12)(34)\rangle. \]
The subgroup \(K\) is normal in \(A_4\): it consists of the identity together with all three double transpositions, and conjugation preserves cycle type.
The subgroup \(H\) has order \(2\). Since \(K\cong C_2\times C_2\) is abelian, every subgroup of \(K\) is normal, so \[ H\trianglelefteq K. \]
But \(H\) is not normal in \(A_4\). For example, \[ (123)(12)(34)(123)^{-1}=(23)(14), \] which is not in \(H\).
Thus \[ H\trianglelefteq K\trianglelefteq G \] does not imply \(H\trianglelefteq G\).