I have the following given equation where I don't understand the conclusion:
Given $$ G^{(n)}N/N = (G/N)^{(n)} = \{N\}$$ Why does it follow, that $G^{(n)} \subseteq N$?
$G^{(n)}$ is the n-th commutator subgroup, $N$ is a soluble normal subgroup and $(G/N)^{(n)}$ is also soluble.
My thoughts/explanation about this is: $$\text{ If we have } G^{(n)}N/N = \{N\}, \text{ then } G^{(n)} \subseteq N, \text{ otherwise there exists } {\sigma \in G^{(n)}\setminus N} \text{ such that } {\sigma N \neq N}, \text{ in contradiction to } G^{(n)}N/N = \{N\}, \text{ since } \sigma N \in G^{(n)}N/N$$