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Is it true that $N$ is a normal subgroup of $G$ iff $\forall g\in G \exists x\in G : gN=Nx$?

I don't think so, but I can't think of a counterexample obviously if $N$ is normal then the above condition is satisfied, since then $x=g$, but how about the other direction?

Thanks.

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1 Answer 1

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Note that $g \in gN = Nx$. But right cosets are a partition of $G$ and $g \in Ng$. Hence $gN = Nx = Ng$ for all $g \in G$ and $N$ is normal.

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