# Proof: symmetric group is not cyclic

Discussion in 'Undergraduate Math' started by Linda Pan, Sep 16, 2003.

1. ### Linda PanGuest

Hi,

I have two ways to prove symmetric group is not cyclic if n is more than 2.

1)
Since symmetric group is not abelian (n>2),
and cyclic group is abelian,
so, symmetric group is not cyclic group (n>2)

2)
Since symmetric group is a permutation group of order n!,
and cyclic group only has order n,
so, when n>2, symmetric group is not cyclic.

P.L.

Linda Pan, Sep 16, 2003

2. ### Arturo MagidinGuest

This one is correct (assuming you have proven that S_n is nonabelian
when n>2).
This one is hopelessly wrong. Why could the symmetric group not be
cyclic of order n!?

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"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
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Arturo Magidin

Arturo Magidin, Sep 16, 2003