There are several proofs of Sylow's three theorems. Historically, Cauchy first proved that

if the order of a symmetric group $S_n$ is $p^aq$ ($p$ is prime, and $q$ is relatively prime to $p$), then $S_n$ has a subgroup of order $p^a$.

Using this theorem, it can be shown every finite group possesses a Sylow-$p$ subgroup (see this).

This shows that Sylow's first theorem for arbitrary finite group can be deduced from Sylow's first theorem for Symmetric group.

Question 1. Are there independent proofs of Sylow's second and third theorems for Symmetric groups, which do not use arguments for general finite group?

Question 2. Can we prove Sylow's second and third theorem for arbitrary group with the information that they are being proved for symmetric groups.

  • $\begingroup$ Maybe with linear algebra $\endgroup$ – Maman Nov 1 '15 at 11:06

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