# Does automorphism preserve the quotient of groups?

Let $N$ be a normal subgroup of $G$ and $\mathrm{Aut}(G)$ be the automorphism group of $G$.

Is there any example that $G/N$ is not isomorphic to $G/\gamma(N)$ for some $\gamma\in \mathrm{Aut}(G)$ ?

• No. non όχι (I have to type a minimal number of characters!) – Bernard Feb 26 '15 at 21:41
• @Bernard: Are you sure ? – mesel Feb 26 '15 at 21:43
• Well, if $\varphi$ is the automorphism, $gN\mapsto \varphi(g)\varphi(N)$ is the isomorphism of $G/N$ onto $G/\varphi(N)$. – Bernard Feb 26 '15 at 21:49
• @Bernard: The problem is that the map is not well defined. (take different repsesentative) – mesel Feb 26 '15 at 21:50
• For me it is well defined: if $g^{-1}g'\in N$, $\varphi(g^{-1})\varphi(g')\in \varphi(N)$. Where is the problem? – Bernard Feb 26 '15 at 21:54

## 1 Answer

$G / N$ is isomorphic to $G / \gamma(N)$ for any automorphism $\gamma$. The reason is that an automorphism $\gamma$ lifts to an isomorphism $\hat \gamma : G / N \rightarrow G / \gamma (N)$ where $\hat \gamma(x N) = \gamma (x) \gamma (N)$.