Sylow intersections

Are halfway tame intersections automatically tame? Are maximal sylow intersections of maximum order?

I thought about asking the following questions about MR215913 on one of those online math help desks:

If P,Q are Sylow p-subgroups of a finite group G, then P∩Q is said to be a tame intersection if the normalizers are also Sylow, that is, if NP(P∩Q) and NQ(P∩Q) are Sylow p-subgroups of NG(P∩Q).

Is it possible for P,Q to be Sylow p-subgroups of a finite group G and NP(P∩Q) to be a Sylow p-subgroup of NG(P∩Q) without NQ(P∩Q) also being a Sylow p-subgroup of NG(P∩Q)?

Often one Sylow p-subgroup, P, is fixed, and the other varies, so there could be two weaker definitions of tame-intersection-with-P. The condition that NP(P∩Q) is a Sylow p-subgroup of NG(P∩Q) has a name and is just a property of the subgroup P∩Q ≤ P and the way G acts on P. However, it seems like the other condition, NQ(P∩Q) is a Sylow p-subgroup of NG(P∩Q), might not be determined by how G acts on P.

A related property does enjoy symmetry: if P is a Sylow p-subgroup of G, and Q is chosen amongst Sylow p-subgroups of G such that P ≠ Q and P∩Q is not properly contained in any other Sylow intersection P ∩ Pg with P ≠ Pg, then we say Q defines a maximal Sylow intersection with P. This is symmetric in that if P defines a maximal Sylow intersection with Q, then Q defines a maximal Sylow intersection with P, and in fact P,Q define a maximal Sylow intersection: P∩Q is not properly contained in any intersection Pg∩Ph with Pg ≠ Ph. Maximal Sylow intersections are tame intersections.

Must a Sylow intersection maximal with respect to subgroup inclusion also be maximal with respect to order?

In other words, do all maximal Sylow intersections with P have the same size?

However, in my efforts to make the question reasonable to answer, I went ahead and answered it:

The group SL(2,23) has Sylow p-subgroup Q16, and has Sylow intersections cyclic of order 4, in all three conjugacy classes. Now one conjugacy class is privileged: it is fully normalized and satisfies the “P” part of the definition of tame-intersection (in any group inducing the same fusion system on P). In smaller groups the unprivileged ones are Sylow intersections and are not at all tame intersections. However, in SL(2,23) there are so many Sylow subgroups that all three classes are intersections, and so they satisfy the “P” part of the definition of tame, but not the “Q” part. So, the answer is just “no”.

I initially expected the answer to be yes, but while I was having trouble proving it, I was also asking GAP about counter-examples, and discovered ΣL(2,25) was a counterexample. I simplified this to SL(2,25) and SL(2,23).

This is particularly cool as yet another example of how fusion systems fail to capture basic information about Sylow intersections. SL(2,7) and SL(2,23) define the same fusion system on Q16, but the subgroups that are Sylow intersections differ.

An even more exciting example is SU(3,5), which has a 2-element Sylow intersection that is both tame and not tame, depending on which Sylows you intersect to get it. In particular, whether or not R ≤ P is a tame intersection, given that it is both fully normalized and a Sylow intersection, is not determined by the fusion system ℱP(G).

I assume I would just get “well the answer is finite” or “it is either yes or no” or some such garbage if I did ask.

The second question is neat, and the answer is also “no”! In this case the counter-example is very mundane: S5 has maximal Sylow intersections ⟨(1,2)⟩ and ⟨(1,2)(3,4),(1,3)(2,4)⟩. This is also the minimal counterexample. I’m not sure what the minimal counter-example for the first question is.