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Why is it that any two distinct subgroups of G of order p (prime) intersect in 1?

It says so here on page 29.

But why is it?

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    Their intersection has to be a subgroup of each of them. What are the only subgroups of a group of prime order?2012-05-06
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    so because the order of $S\cap P$ for $P, S \in Syl_pG$ and $S\neq P$ divides the order of $S$ and the order of $P$ it must be either 1 or $p$ but it can't be $p$ because $P$ and $S$ are distinct?2012-05-06
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    Exactly. So their intersection must be trivial.2012-05-06
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    great. thanks a lot!2012-05-06
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    @lhf: Done. $\quad$2012-05-06

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Converting my comment to an answer to get the question off the Unanswered list:

HINT: If $H$ and $K$ are subgroups of $G$, $H\cap K$ is a subgroup of $H$ and of $K$. What are the only subgroups of a group of prime order?