Talk:Sylow theorem

I removed this: From these observations, we can derive the following useful formula relating a group's order and its Sylow subgroups: which preceded the last statement. I don't think this last statement is a mere consequence of the first statements; it requires its own independent proof. AxelBoldt 18:04 Nov 6, 2002 (UTC)

Yep. I was really thinking about highlighting the "meat" (i.e., the hard thing to remember exactly) about n_p. I separated these out and gave some examples. Chas zzz brown 19:36 Nov 6, 2002 (UTC)

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Nitpick: The theorem is still true if p does not divide |G|; it's just that a Sylow p-subgroup then has order p⁰ = 1. Chas zzz brown 04:55 Nov 12, 2002 (UTC)

If we count the trivial group as a p group, and I don't know if that is common, or a good idea. The theorem that every p group has non-trivial center is then false. AxelBoldt 05:40 Nov 12, 2002 (UTC)

Good point :) Chas zzz brown 05:41 Nov 12, 2002 (UTC)

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Added proofs of Sylow theorem based on W. R. Scott's Group Theory, Dover publications.

These proofs rotate more around the idea of conjugacy classes, normalizer, and centralizers; rather than the orbits and stabilizers from the concept of the group action of inner homomorphisms of G on G. I don't know if it's the current "standard" proof; Scott's book is from the 1960s. Hopefully the proof is not overly prolix. :) Chas zzz brown 10:15 Nov 12, 2002 (UTC)