TITLE: Sylow's theorem 3rd Proof  Page 96 I.N.Herstein
QUESTION [5 upvotes]: I was just going through the 3rd Proof of Sylow's theorem given in the "Topics In Algebra" Book by I.N. Herstein. It looked very interesting and i really liked its Philosophy. My question what is its significance, and how can it be applied to problems, or something else.

REPLY [14 votes]: This is the proof that uses the lemma that if a finite group $G$ has
a Sylow $p$-subgroup then so does each subgroup of $G$. To complete
the proof of existence of Sylow $p$-subgroups, it suffices to show
one can embed each group in a group with a Sylow $p$-subgroup.
By Cayley's theorem each finite $G$ embeds in $S_n$ with $n=|G|$
and $S_n$ embeds in $S_{p^k}$ where $p^k\ge n$. One then writes
down a Sylow $p$-subgroup of $S_{p^k}$ (essentially an iterated
wreath product of $C_p$s).
But a slicker conclusion is to embed $S_n$ in $GL_n(p)$
(via permutation matrices),
as one sees with little effort that the upper triangular matrices
with $1$s on the diagonal form a Sylow $p$-subgroup of $GL_n(p)$.