Sylow Theorems are very powerful in finite group theory.

Two natural questions come to mind:

1) Given a finite group $G$ and a $p$-subgroup $H$ of $G$, how many Sylow $p$-subgroups of $G$ contain $H$?

2) Let $P$ be a Sylow $p$-subgroup of a finite group $G$ and $k$ be a fixed positive integer. How many subgroups of $G$ of order $p^k$ are contained in $P$?

Does anybody know of if there are some results which address the above two questions?


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    $\begingroup$ Your second question does not seem to be about Sylow theory, but rather a question about an arbitrary finite $p$-group $P$, in which case there may not be a lot one can say. The answer could be as few as $1$ (in the case of a cyclic group), or as many as $\log_p \left| P\right|-1$ (take $P$ to be elementary). But perhaps the first question could lead to something interesting. $\endgroup$ – James Sep 7 '14 at 7:35
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    $\begingroup$ For the first question, you can show that the number is $\equiv 1 \mod {p} $ $\endgroup$ – spin Sep 7 '14 at 8:38
  • $\begingroup$ @spin Right! Let $A$ be a Sylow $p$-subgroup of $H$. Let $A$ act on $\text{Syl}_P(G)$ by conjugation. Then $Q$ is a fixed point of this action if and only if $A\subseteq Q$. From here we get the result. Thanks! $\endgroup$ – caffeinemachine Sep 7 '14 at 9:49

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