I've been taught that the first Sylow theorem states:
If $|G|=n$ and $p^km=n$ for $p$ prime and $p\not\mid m$, then $|G|$ has a subgroup of order $p^k$.
However, this website states the theorem differently:
http://mathonline.wikidot.com/the-sylow-theorems
Let $G$ be a finite group. If $p$ is a prime, $k\geq0$, and $p^k$ is a divisor of $|G|$ then $G$ has a subgroup of order $p^k$.
From this definition, it sounds like any prime power that divides $|G|$ has a subgroup of that order, which is different statement than the one my textbook gives. I've read through the proof on the Math Online and couldn't see anything wrong about it, yet I still have doubts… Thanks in advance.
Edit: So both statements are true, and I see now that the MathOnline one is stronger. But when I look up "Sylow's First Theorem" on the Internet I tend to find proofs of the weaker statement. I wonder why it's the weaker statement that gets presented everywhere?
Best Answer
Mathonline version is strictly stronger than the first one, so these statements are not equivalent. However, these two statements are both right.