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There is indeed a strong analogy between the study of $p$-local subgroups and the theory of buildings, at least for groups of Lie type. More precisely, if $G$ is a finite group of Lie type over a fiel …

The map $T \colon \mathrm{PSL}_2(p) \to \operatorname{Sym}(\mathbb{F}_{2^n}) \colon f \mapsto T_f$ does not have its image in $\mathrm{GL}_n(2)$ for other Mersenne primes $p = 2^n - 1$, unlike the cas …

As indicated in Martin Seysen's comment, this construction can be found in Carter's book "Simple Groups of Lie type". More precisely, this is Proposition 13.6.3, and your "naive approach" is indeed ex …

The following theorem (which does not take the order $N$ of the group $G$ into account) shows that all possible combinations of $a$, $b$ and the order of $xy$ are possible. See Theorem 1.64 from Milne …

It is not very hard to see that for each prime power $q$ and natural numbers $n,h$, we have an embedding $$\iota \colon \mathrm{GL}(n,q^h) \hookrightarrow \mathrm{GL}(nh, q),$$ obtained by choosing a …

I think it's worth adding that there is a very detailed analysis of the orthogonal groups over arbitrary fields (not just finite fields, and including characteristic 2) in Dieudonné's "La Géométrie de …

The group $G = C_2 \times C_2 \times C_3 \times C_3$ has this property (for $p=3$).

In his answer to this previous MO question, Gjergji Zaimi referred to the statement that for every finite group $G$ of order $n$, there is a bijection $\sigma \colon G \to \mathbb{Z}/n\mathbb{Z}$ sati …

I think that you can find the formulas that you are looking for in the paper "An arithmetic method of counting the subgroups of a finite abelian group" by Marius Tarnauceanu, Bull. Math. Soc. Sci. Mat …

In the case of groups of rank 2, such as your examples $\mathrm{SL}_3(\mathbb{F}_2)$ or $\mathsf{G}_2(3)$, the building is rather easy to describe (either as an incidence geometry or as a bipartite gr …

Somewhat related to Igor Pak's comment is the classification of the finite irreducible Coxeter groups. Of course they are not "simple" as groups, but the irreducibility seems the natural replacement f …

I hope that my question is appropriate for MO, since it might turn out te be mainly a question about GAP or other group theory software. Is there an algorithm to produce all non-nilpotent groups o …

Yes, $G$ always contains a cyclic subgroup of composite order. Note that all Sylow subgroups of $G$ are cyclic, i.e. $G$ is a Zassenhaus metacyclic group. Such groups have a very precise structure: th …

The former group can be seen as the group of unitriangular $3 \times 3$-matrices over the field with $p$ elements: $$G = \left\{ \begin{pmatrix} 1 & * & * \\ 0 & 1 & * \\ 0 & 0 & 1 \end{pmatrix} \righ …

You were almost there: Since $N$ and $F(G)$ are two normal subgroups intersecting trivially, they commute. But now take a non-trivial element $a \in A \cap F(G)$; then by the previous observation, $a$ …