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.



Sylow's theorem 3rd Proof  Page 96 I.N.Herstein
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 $GL\_n(p)$.


