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Q:

check if an automorphism is the identity is not polynomial

How to check if a automorphism $\sigma : G/N \to G/N$, $N \triangleleft G$ is the identity. I know a theorem saying that
$$
\operatorname{Ker}(\sigma-\mathrm{id})=\operatorname{Ker}(\sigma)N/N ec5d62056f darfrod
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