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.

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Page 96 is "Ring Theory" in my Herstein. –  darij grinberg Aug 1 '10 at 15:39
thats a problem right. i cant help. The proof is using symmetric groups. –  Chandrasekhar Aug 1 '10 at 15:41
At least you could tell us what edition of Herstein you are using, and whether the page number 96 refers to the actual book, a PDF, a DJVU, or whatever –  darij grinberg Aug 1 '10 at 15:46
OK.. its better than you see that 3rd proof in sylow theory. I am using second edition. –  Chandrasekhar Aug 1 '10 at 15:48
Alas, I can only find the first edition. But it wouldn't harm if you would post the basic ideas of the proof here. –  darij grinberg Aug 1 '10 at 16:09

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 $GL_n(p)$.

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