Mikko Korhonen
  • Member for 11 years, 3 months
  • Last seen this week
$\operatorname{PSL}(2,\mathbb{F}_p) $ does not embed in $\mathfrak{S}_p$ for $p>11$
10 votes

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 ...

View answer
What are the automorphism groups of direct products of dihedral group D4
10 votes

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. ...

View answer
Order of products of elements in symmetric groups
Accepted answer
9 votes

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 ...

View answer
For which $n$ can $S_n$ act transitively on $n+k$ elements?
9 votes

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] =...

View answer
A property forcing the Frobenius-Schur indicator to be positive
Accepted answer
9 votes

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 ...

View answer
Fundamental representations and weight space dimension
Accepted answer
9 votes

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 ...

View answer
Number of involutions in a finite group modulo $4$
Accepted answer
9 votes

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–...

View answer
Applications of Frobenius theorem and conjecture
8 votes

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 ...

View answer
Finite groups with the same character table *including* class types, and square-free order
Accepted answer
7 votes

(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 ...

View answer
Finite groups with few conjugacy classes of maximal subgroups
7 votes

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 ($G$ ...

View answer
Schreier conjecture -- without a simple proof? and sporadic simple groups
5 votes

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: ...

View answer
Formula for the Frobenius-Schur indicator of a finite group?
Accepted answer
5 votes

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 ...

View answer
Are old issues of math journals from Belarus Academy of Sciences available online?
3 votes

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 ...

View answer
Subgroups of algebraic groups containing regular unipotent elements
Accepted answer
3 votes

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 ...

View answer
For a fixed dominant weight $\lambda$, are almost all dominant weights in the same coset above it?
3 votes

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} \...

View answer
Reference Request: Derived group of $\mathscr R_u(B)$
Accepted answer
3 votes

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}...

View answer
Conjugacy of elements in a parabolic subgroup
2 votes

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 ...

View answer
Lower bounds on the number of elements in Sylow subgroups
2 votes

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 +...

View answer