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

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, $a$ …

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 …

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 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 …

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 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 …

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 …

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 …

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 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 …

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 …

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 …

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 …