Nilpotent systems of subgroups in GAP

Question

The following is a cross-post from the GAP forum.

I have formulated and proved a certain proposition regarding collections of nilpotent subgroups of a finite group. If $G$ is a finite group, then I will call $\mathfrak{F}_G$ a system of nilpotent subgroups for $G$ if $\mathfrak{F}_G$ consists solely of nilpotent subgroups of $G$, and further satisfies:

1. $1\in \mathfrak{F}_G$, and $G \notin \mathfrak{F}_G$;
2. if $K \in \mathfrak{F}_G$ and $J \leq K$, then $J \in \mathfrak{F}_G$;
3. if $K \in \mathfrak{F}_G$, then $K^g \in \mathfrak{F}_G$ for all $g \in G$;
4. for each subgroup $K$ of $G$ (in particular, for $G$ itself), the maximal elements of the sub-poset $\mathfrak{F}_K := \{S \in \mathfrak{F}_G : S \leq K\}$ (with inclusion as partial order) forms a single $K$-conjugacy class $\mathcal{M}_K$;
5. for any two subgroups $K$, $L$ of $G$ with $K \leq L$, we have that $(K:S)$ divides $(L:T)$, where $S \in \mathcal{M}_K$, and $T \in \mathcal{M}_L$.

As an example, if $G$ is not a $p$-group, and $p$ divides $|G|$, for some fixed prime $p$, then we can take $\mathfrak{F}_G$ to be the collection of all $p$-subgroups of $G$. Then Sylow's theorems guarantee that $\mathfrak{F}_G$ is a system of nilpotent subgroups for $G$.

In fact, there is reason to speculate that if $\mathfrak{F}_G$ is to be a system of nilpotent subgroups for a group G, then conditions 1.--5. are together strong enough for $\mathfrak{F}_G$ to be some "well-known" family of subgroups (like Sylow), but I haven't been able to prove anything like that.

I would kindly ask for some help in formulating a GAP search for such systems of nilpotent subgroups, and will start a bounty as soon as possible.

Answer 1

If I recall correctly, a finite group which is not nilpotent has a maximum (normal) nilpotent subgroup, which is also known as the Fitting subgroup: https://en.wikipedia.org/wiki/Fitting_subgroup

A fairly easy way to find all nilpotent subgroups in GAP is to just use a for loop that runs of the set of all subgroups. I constructed a similar loop for the purposes of searching for groups isomorphic to their own automorphism group (and I found all such groups of order less than 512 in a couple hours of work).

Edit: Once you have the collection of nilpotent groups, refining it down to just those that satisfy the properties required to be the system of interest can all be done with appropriately constructed loops and basic GAP commands that can be found in the reference materials (section 39 of reference manual is where they can be found if memory serves).

When I went looking for some info related to nilpotent groups, I came across the idea of a fusion system for a group (this: http://web.mat.bham.ac.uk/D.A.Craven/docs/lectures/fusionsystems.pdf), which just might be what you're looking at here.