I was told yesterday that in a system of Sylow $p$-subgroups of a finite group $G$, if, $\{S_1,S_2, \cdots, S_n\}$ make up the system, it can happen that, say, the intersection of $S_1$ and $S_2$ has order $p^k$, while the intersection of $S_3$ and $S_4$ has size $p^j$, with $k \neq j$.

1) My first question is if some examples are available of this. I think I can't find examples in the alternating groups and symmetric groups, due to multiple transitivity.

2a) If $j = k$ in the above, must the intersections $S_1 \cap S_2$ and $S_3 \cap S_4$ be isomorphic?

2b) If those two intersections in 2a) above are isomorphic, then must there be an inner automorphism mapping $S_1$ to $S_3$ and $S_2$ to $S_4$? That is, must Sylow systems have as much conjugacy as "possible", in addition to simple transitivity?


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    $\begingroup$ You should be able to find counterexamples for all claims by choosing the right groups $G$ and $H$ and looking at $G\times H$. The Sylow subgroups of $G\times H$ are the direct products of the Sylows of $G$ with those of $H$, so just pick the latter ones for your needs. For 1) and 2b) you can take $p=2$ and $G=H=S_3$. For 2a) $p=2$, $G=Z_5\rtimes Aut(Z_5)$ and $H=A_5$ (or another group with 2-Sylows isomorphic to $Z_2\times Z_2$ and two 2-Sylows intersecting trivially). $\endgroup$ – j.p. Apr 14 '15 at 6:18
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    $\begingroup$ In the alternating group $A_7$ the $3$-Sylow subgroups $S_1=S_3=\langle (123), (456)\rangle$, $S_2=\langle (123), (457)\rangle$ and $S_4=\langle (124), (356)\rangle$ answer 1). For 2) the examples would become biggish, but should exist. $\endgroup$ – j.p. Apr 14 '15 at 7:51

Let $C_3$ be the cyclic group of order $3$ and $D_8$ the dihedral group of order $8$, and let $G$ be the regular wreath product $C_3 \wr D_8$, which has order $3^8 \cdot 8$.

There are $3^7 = 2187$ Sylow $2$-subgroups of $G$, and all subgroups of $P$ arise as intersections of two of these. Since $P$ has non-isomorphic subgroups of order $4$, this answers Questions 1 and 2a) in the negative.

I found examples of Sylow $2$-subgroups $P_1,P_2,P_3$ with $|P_1 \cap P_2| = |P_1 \cap P_3| = 2$, such that $P_2$ and $P_3$ are not conuugate in $N_G(P_1)$, so the answer to 2b) is also no.


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