Let $k$ be the largest positive integer such that $p^k \mid |G|$ and let $|H|=p^i$ where $0<i < k$. What should I prove to say that $H \subset H_1$ where $|H_1|=p^{i+1}$, and what guarantees the existence of $H_1$?
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We have had such a question here: http://math.stackexchange.com/questions/150534/show-a-certain-group-is-contained-in-a-sylow-p-group – Cocopuffs Apr 13 '13 at 19:24
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1@Cocopuffs Thank you but,I'm not looking for a solution, I'm trying to find a way to continue my method. – user10444 Apr 13 '13 at 19:27
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I think you want $0\le i\lt k$, not $0\lt i\le k$. – MJD Apr 13 '13 at 21:14
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@MJD yes that is true, I will edit it. – user10444 Apr 13 '13 at 21:15
1 Answers
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Some hints from J.D.Dixon. I found them for you:
- Write the order of $G$ as $|G|=|Px_1H|+|Px_2H|+...+|Px_sH|$ for some elements $x_i\in G$.
- If $|P|=p^r$ then $p^{r+1}$ does not divide $|G|$.
- So there is some double coset $PxH$ that $p^{r+1}$ does not divide its order.
- Use the point that $|PxH|=\frac{|P|}{[H:x^{-1}Px\cap H]}$
- $H$ is a $p-$ group.
Mikasa
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This is a bit too advanced for me, as I have never worked with double cosets. I'll try to read about them on Wikipedia and get back to these hints. Thank you – user10444 Apr 13 '13 at 19:36
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How are you?;-) I hope you slept blissfully! I will likely head to bed soon (almost midnight for me!) – amWhy Apr 14 '13 at 04:57
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@amWhy: So hurry. there is a warm and charming bed prepared for you. I slept well, and I hop you will have the same full of stars. ;-) – Mikasa Apr 14 '13 at 05:03
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@amWhy: My tel line has not been implemented yet. I miss being in the site, with you, with others, doing Maths and.... – Mikasa Apr 15 '13 at 07:54