Let $\lambda(n) = \lambda(\mathbb{Z}_n^*)$ be the Carmichael function and $\phi(n) = | \mathbb{Z}_{n}^{*} |$ be the Euler function. I need to show:
$$\mathbb{Z}_{\lambda(n)}^* \le \mathbb{Z}_{\phi(n)}^*$$
But why should $\mathbb{Z}_{\lambda(n)}^* \le \mathbb{Z}_{\phi(n)}^*$ one be a subgroup of the other? This was stated in one course on cryptography but I cannot find a good justification for it.
The claim was wrong
In general this only holds for the subset relation. Still it would be great to find a counterexample. In return, I leave here an argument why the subset relation holds.
