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?
Thanks.
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.