Originally Posted by

**ThePerfectHacker** For anyone who has John Fraleigh "A First Course in Abstract Algebra" 7/e would be easier to follow. (I definitely know some of you have it, because many people referred to it already, very popular book).

Let $\displaystyle p$ be a prime.

On Page, 324-325

John is talking about the first Sylow theorem.

"Given a finite group $\displaystyle |G|=p^n m$ with $\displaystyle n\geq 1$ and $\displaystyle p\not | m$, then there exists a subgroup of order $\displaystyle p^{i}$ for $\displaystyle 0\leq i\leq n$.

Furthermore, each subgroup of order $\displaystyle p^i$ is a normal subgroup of order $\displaystyle p^{i+1}$. For $\displaystyle 0\leq i<n$.

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The first part of the proof I understand, the second part ails my soul and makes my blood cold and so by degrees.

I believe that John wanted to write,

$\displaystyle p^i$ is a normal subgroup of **some** group of order $\displaystyle p^{i+1}$. He makes it appear as though it is true for all subgroups, which is unlikely.

Anyone know what I am talking about?