# Invariance under a normal subgroup

Let $$G$$ be a group acting on a set $$A$$. Let $$N$$ be a non-trivial normal subgroup of $$G$$. Suppose that $$S$$ is an $$N$$-invariant set, i.e. $$n \cdot s \in S$$ for all $$s \in S$$, $$n \in N$$. Must $$S$$ be $$G$$-invariant?

I suspect the answer is yes (for example, it is true in the situation of a Galois group acting on a Galois field extension). I've been playing around with it for a while and haven't been able to get anything.

• Sorry N-invariant means N(S)=S? – mathpadawan Dec 6 '18 at 23:36
• @mathnoob Yes, I edited it for clarity. – vukov Dec 6 '18 at 23:37
• What if $N$ is the trivial normal subgroup $\{e\}$? – Catalin Zara Dec 6 '18 at 23:39
• @CatalinZara You're right, I meant to require non-trivial. – vukov Dec 6 '18 at 23:42

Let $$G$$ be a group and $$N$$ a normal subgroup of $$G$$. Let $$A=G/N$$ be the collection of left cosets with the left multiplication action, and let $$S=\{eN\}$$. Then letting $$G\cdot T=\{g\cdot t\mid g\in G, t\in T\}$$ for any $$T\subseteq A$$, we have in particular $$N\cdot S=S$$ but $$G\cdot S=G/N\neq S$$.