The bookUnit 2 · Groups40 / 83

Permutation groups: cycles, sign, conjugacy in S_n and A_n

Why this is asked: Cycle type determines conjugacy in Sₙ (but splits in Aₙ). Count elements of a given order by counting cycle types; know the order of a permutation is the lcm of its cycle lengths.

Permutation groups — cycle type, sign and conjugacy

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See it move

Cycle type decides everything in Sₙinteractive

Order, sign and conjugacy all read off one decomposition — plus where the rule changes in Aₙ.

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The trap here

“ $A_{n}$ is simple for every n ≥ 3” — false

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