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Although some fiddling will be necessary to get a precise result, one can understand problems of this type without using powerful modern results in group theory. For large enough $n$, there are a lot …

You might find useful the paper of Carlo Casolo, "Some linear actions of finite groups with $q^\prime$-orbits". If I understand correctly, this handles nonsolvable groups under the additional assumpt …

Let $P$ be a Sylow $3$-subgroup of $G$ (which we may assume to be nontrivial) and consider the conjugation action of $P$ on the set $X$ of all elements of order six in $G$. Remove from $X$ all of its …

I think that with some work you can characterize the examples where $G$ is solvable as follows. (I did not check carefully, so you can take this with a grain of salt.) If $H$ has trivial core in $G$ …

Here are infinitely many examples showing that the answer to question 1 is negative. Take $n \equiv 1 \bmod 4$ with $n>5$ and let $\chi$ be the character of the symmetric group $S_n$ associated to th …

Let $p$ be the smallest prime divisor of $n$. I believe that $f(n)=n/p+1$. The key points are as follows. 1) If $H \leq S_n$ is primitive and contains a $2$-cycle, then $H=S_n$. 2) If $1<k \leq m< …

Such triples exist, I think. First, embed $PGL_2(p)$ in $S_{p+1}$ through its action on $1$-spaces from ${\mathbf F}_p^2$. This maps elements of order $p+1$ in $PGL_2(p)$ to $(p+1)$-cycles, and el …

I believe that it was already known to Jordan that if $n \geq 8$ and $G$ is a primitive subgroup of $S_n$ containing a double transposition, then $G$ contains $A_n$. As a subgroup of $S_p$ containing …

Maybe I misunderstand the question. I interpret it as follows: you have a finite group $G$ and some (minimal) set $S$ of generators for $G$ that satisfies the relations defining a Coxeter group $W$ w …

Let $G=S_n$, acting on the ${{n} \choose {2}}$ $2$-sets from $[n]$. The stabilizer of a $2$-set $X$ is maximal in $S_n$ and has two orbits on the remaining $2$-sets, one consisting of those that inte …

Please allow me to substitute as follows in order to avoid typos: $r_{11}=a$, $r_{12}=b$, $r_{21}=c$, $r_{22}=d$, $t_{21}=e$, $t_{22}=f$. Upon examining the character tables of $S_4$, $A_4$, and $D_8$ …

Here is a proof by contradiction. Assume an $8$-tuple $E$ from $G$ satisfies all of your relations. Let $\Phi=\langle w^2,x,y \rangle \leq G$. It is straightforward to observe that (A) $\Phi$ is abel …

Here is another infinite class of examples: Say $q>3$ is a prime power and $a$ is a square-free odd integer. Then $L_2(q)$ embeds in $L_2(q^a)$, and the lattice of subgroups above the image of such a …

Maybe it is difficult to say what is strictly "outside" of finite group theory. However, to add to Igor's answers, I can suggest that you look at the work of Bob Guralnick and various collaborators. …

Say $G=S_n$ and $H$ is a Young subgroup with three orbits, no two of which have the same size and no two of which have sizes summing to $n/2$. Then the only subgroups between $H$ and $G$ should be th …