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I was working on a problem asking to show that a group of order $132=2^2*3*11$ is not simple. The solution basically involves counting up elements and deciding how many are in a Sylow $2$-subgroup, $3$-subgroup, or $11$-subgroup. In the solution, we use the fact that the intersection between all of these subgroups is trivial. Is this always the case? Under what circumstances is the intersection of two Sylow $p$-subgroups trivial?

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    $\begingroup$ not simple means that it has nontrivial normal subgroups $\endgroup$
    – janmarqz
    Commented Dec 15, 2016 at 22:11
  • $\begingroup$ The intersection of two different Sylow $p$-subgroups is certainly trivial if the order of a Sylow $p$-subgroup is $p$. For a cyclic group of prime order has only the two obvious subgroups. If the order of Sylow $p$-subgroups is $p^k$ with $k > 1$, without information about the group, a priori the intersection can have any order $p^r$ where $0 \leqslant r < k$. $\endgroup$ Commented Dec 15, 2016 at 22:17
  • $\begingroup$ If the intersection $T$ of two $p$-Sylows is nontrivial, then you can take w.l.o.g. $T$ to be maximal among those intersections and look its normalizer $N :=N_G(T)$ and centralizer $C_G(T)$. For any $p$-Sylow of $G$ containing $P$ you have $P<N_P(G)$, so $N$ has at least two $p$-Sylows and it often helps to apply Sylow's theorems to the normalizer $N$. For looking at the centralizer a theorem of Burnside about $p$-complements will be helpful. $\endgroup$
    – j.p.
    Commented Dec 16, 2016 at 10:03

1 Answer 1

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Let $p$ be a fixed prime dividing the order of $G$. Suppose $H_1,H_2,\dots,H_n$ are the $p$-Sylow subgroups of $G$. Then $H=H_1\cap H_2\cap\dots\cap H_n$ is normal in $G$, because $p$-Sylow subgroups are conjugate of each other. If $H$ is not trivial, then $G$ is not simple.

So, in order to show $G$ is not simple, you can assume that, for each $p$ dividing $|G|$, the $p$-Sylow subgroups intersect trivially. Note that, in general, the intersection of all the $p$-Sylow subgroups is not trivial: just consider a $p$-group or, more generally, a group having a unique $p$-Sylow subgroup.

Of course the intersection of a $p$-Sylow and a $q$-Sylow subgroups is trivial when $p$ and $q$ are distinct.

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