Counterexample

Is this true?

$A_{n}$ is simple for every n ≥ 3

No — it is false.

The counterexample

$A_{4}$ has the normal subgroup $V_{4} =$ {e, (12)(34), (13)(24), (14)(23)}

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Execution slip

The idea was right. The computation was not.

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groups permutations

Go deeper· Explained in full in Permutation groups — cycle type, sign and conjugacy· Asked in CSIR NET December 2024· From Groups › Permutation groups: cycles, sign, conjugacy in S_n and A_n

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.

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