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As $p^2+1 \equiv 2 \bmod 4$, a Sylow $2$-subgroup of $G$ has order four. As $p^2+1 \equiv 2 \bmod 3$ and $p>3$, $G$ has no element of order three. It follows now that a Sylow $2$-subgroup $X$ of $G$ …

An extraspecial $p$-group of exponent $p$ contains exactly three characteristic subgroups, $1$, $G$ and the center of $G$. Let $Z$ be the center of $G$ (so $Z=[G,G]=\Phi(G)$). The elementary abelian …

Here is an answer using character theory. For an irreducible character $\chi$ of a finite group $G$, define $\nu_2(\chi)$ to be $1$ if $\chi$ is afforded by a real representation, $0$ if $\chi$ is no …

How about if we follow Derek Holt's beautiful argument until we establish the following two facts? 1) $G$ has 10 Sylow 3-subgroups. 2) Let $P_3$ be a Sylow 3-subgroup of $G$. In the action of $G$ o …

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 …

There are pairs G,H of nonisomorphic p-groups with isomorphic subgroup lattices (and therefore of the same order). The book ``Subgroup Lattices of Groups", by R. Schmidt, is an excellent reference on …

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 …

As $S_n$ also embeds in $GL_{n-1}({\mathbb F}_2)$, you only need to check that $n!$ is not divisible by $2^n$.

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 …