Sylow's theorem 3rd Proof  Page 96 I.N.Herstein 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.
 A: 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 $GL_n(p)$.
A: As Robin Chapman has given an elegant and compact proof, I content myself with answering your query about applications. Sylow's three theorems are a very interesting tool to classify groups of low cardinality. Some exercises are on this given in Herstein's book itself. Upto groups of order 60, you can use just the three theorems of Sylow and classify them as direct or semi-direct products. Here all three theorems are needed; only the third proof of Herstein proves all three.
The case of groups of order 60 is a bit intricate; the appropriate reference is M. Artin's Algebra book.
