For n > 1, the group A_{n} is a normal subgroup of the symmetric group S_{n} with index 2 and has therefore n!/2 elements. It is the kernel of the signature group homomorphism sgn : S_{n} → {1, 1} explained under symmetric group.
The group A_{n} is abelian iff n ≤ 3 and simple iff n = 3 or n ≥ 5. A_{5} is the smallest nonabelian simple group.
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