Groups | Part IA, 2008

Define the signature ϵ(σ)\epsilon(\sigma) of a permutation σSn\sigma \in S_{n}, and show that the map ϵ:Sn{1,1}\epsilon: S_{n} \rightarrow\{-1,1\} is a homomorphism.

Define the alternating group AnA_{n}, and prove that it is a subgroup of SnS_{n}. Is AnA_{n} a normal subgroup of SnS_{n} ? Justify your answer.

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