Here is the 1868 paper of Jordan mentioned in the comments: C. Jordan. Note sur les équations modulaires, C. R. Acad. Sci. Paris 66 (1868), 308-312. Here is Jordan's proof from that paper, as I ...

The following papers are relevant: [1] J. N. S. Bidwell, M. J. Curran, and D. J. McCaughan, Automorphisms of direct products of finite groups, Arch. Math. 86, 481 – 489 (2006). [2] J. N. S. ...

The main theorem in a paper of G. A. Miller [1] is the following: THEOREM. If $l, m, n$ are any three integers greater than unity, of which we call the greatest $k$, it is always possible to find ...

Fix $k > 0$. Suppose that $n > 6$ and $\frac{n(n-3)}{2} > k$. If $[S_n : H] \leq n+k$, then $H$ is one of the following: $S_n$, $A_n$, $S_{n-1}$, or $A_{n-1}$. So in particular if $[S_n : H] =...

Here is a more general statement, see also Lemma 1.2 in [1]. Lemma: Let $Z$ be a self-dual $kG$-module which admits a non-degenerate $G$-invariant symmetric (alternating) bilinear form $b$. Suppose ...

Let $\mathfrak{g}$ be a simple Lie algebra over an algebraically closed field of characteristic $0$ and denote the fundamental highest weights by $\varpi_1, \ldots, \varpi_l$, where $l$ is the rank of ...

EDIT: Actually, there is an older reference which will give the result below. See Herzog, Marcel Counting group elements of order $p$ modulo $p^2$. Proc. Amer. Math. Soc. 66 (1977), no. 2, 247–...

Since nobody gave any examples of applications of Frobenius conjecture, here's a small one I read about recently. We will only consider finite groups in what follows. For a group $G$, denote the set ...

(Turning my comments into an answer). A finite group with all Sylow subgroups cyclic is called a $Z$-group. According to review MR0470050 in MathSciNet, in [1] it is shown that a $Z$-group is ...

See Kano, Mikio. On the number of conjugate classes of maximal subgroups in finite groups. Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 7, 264--265. link for proofs of the following ($G$ ...

Regarding your second question, for every finite simple group the structure of its outer automorphism group is known. For the sporadic groups, there is the following 2011 preprint on arXiv: ...

For one answer, here is a theorem due to Thompson and Willems (Bilinear forms in characteristic $p$ and the Frobenius-Schur indicator, Lecture Notes in Mathematics 1185, pg. 221-230). For an ...

It seems that for older issues of these journals, there are no scanned copies available online. But depending on your location, a nearby university library might be helpful for getting scans of ...

I do not know a reference, but I have thought about the same question. Here is a sketch using arguments that are in the literature. For some basics about regular unipotent elements, see for example ...

To me the answer seems to be yes. Let $\varpi_i$ be the $i$th fundamental dominant weight. Recall first that since $\Phi$ is irreducible, for all $i$ we have $\varpi_i = \sum_{j = 1}^l q_{ji} \...

I think the answer is yes. We can assume that $G$ is semisimple. As you have seen already, and which is is clear from the commutation relations, we have $[U, U] \leq \prod_{\alpha \in \Phi^+ - \Delta}...

I think your question has a negative answer. For reductive $G$, there are only finitely many conjugacy classes of unipotent elements in $G$. However, for a parabolic subgroup $P < G$ with ...

This is just a small partial result and some comments. I think the following should settle the case $k = 2$. Suppose that $G$ is a group with Sylow $p$-subgroups of order $p^n$ and that $n_p(G) = 2p +...