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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 …

For any finite group $G$, let $$\theta(G) := \sum_{g \in G} \frac{o(g)}{\phi(o(g))},$$ where $o(g)$ denotes the order of the element $g$ in $G$, and where $\phi$ is the Euler totient function. It is …

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 …

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 …

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 permutation groups $A = PSL(2,7)$ with its natural action on the projective line $\mathbb{P}^1(\mathbb{F}_7)$ and $B = A\Gamma L(1,8)$ with its natural action on the affine line $\mathbb{F}_8$ hav …

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 …

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 …

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 …

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 …

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 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 …

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 …

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

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 …