12 votes

The degree of a constant polynomial on a finite group

If $X = A_5$ then $\exp(X) = \mathop{\rm lcm}(2,3,5) = 30$ but $\deg 1_X \leq 10$. Indeed if $c \in X$ is a 5-cycle then $$ f(x) := x c x c^2 x c^3 x c^4 x $$ satisfies $f(x)^2 = 1$ for all $x \in X$ ...
10 votes
Accepted

The number of polynomials on a finite group, II

This is an answer to problem 1 and problem 3: As noted in the comments, $\operatorname{Poly}(G)$ with pointwise multiplication is a group for finite $G$. Consider the homomorphism $\operatorname{Poly}(...
  • 6,189
8 votes

Splitting of a finite group with no abelian subfactor in composition series

There are groups that look like wreath products, but where the base group has no complement, so they are not semidirect products. The theory is described in an old paper of mine (and probably ...
  • 34.4k
7 votes
Accepted

Presentation of the Monster as a Hurwitz group

I have computed two pairs of generators $(a,b)$ of the Monster satisfying the relations $a^2 = b^3 = (ab)^7 = 1$ using [1]. In both cases $a$ is of class 2B, $b$ is of class 3B in the Monster, and ...
7 votes
Accepted

Extensions of a simple group by an elementary abelian $p$-group

I think this can often fail even for the non-trivial representation $V$ of smallest possible dimension. For a reductive group $G$ over $\mathbf{Z}_p$, the group $G(\mathbf{Z}/p^2 \mathbf{Z})$ will ...
7 votes
Accepted

How to make Burnside's formula compatible with point counting for varieties over finite fields?

The issue is that $X(\mathbb{F}_p)/G$ is not the same thing as $(X/G)(\mathbb{F}_p)$. A simpler example is to take $p$ odd, $X = \mathbb{A}^1$ and let $S_2$ act by $\pm 1$. There are $\tfrac{p+1}{2}$ ...
6 votes

"Novelty" maximal subgroups in $S_n$

The best reference for this topic seems to be this paper: A classification of the maximal subgroups of the finite alternating and symmetric groups, Martin W Liebeck, Cheryl E Praeger, Jan Saxl, ...
  • 34.4k
5 votes
Accepted

Length of representation of $GL_n(\mathbb{F}_q)$ in functions on Grassmannian

I believe that this is answered in Proposition 5.1 of this paper, which says that it is $\text{min}(k,n-k) + 1$.
  • 40.4k
3 votes

Splitting of a finite group with no abelian subfactor in composition series

The group $\operatorname{Aut}(A_6)$ has a unique simple normal subgroup, $\operatorname{Inn}(A_6) = A_6$. It is a nonsplit extension $$1 \rightarrow A_6 \rightarrow \operatorname{Aut}(A_6) \rightarrow ...
  • 2,546
3 votes

Prime divisors of nonabelian simple group and of its outer automorphism group

Since outer automorphism groups of finite simple groups of Lie type are rather small solvable groups, and outer automorphisms are products of graph, field, and diagonal automorphisms in general, it ...
1 vote
Accepted

What are the maximal closed subgroups of $ \operatorname{SU}_3 $?

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\Cl{Cl}$Yes the the above is the correct list of maximal closed subgroups of $ \SU_3 $. Antoneli, ...
1 vote
Accepted

Fusing the $\mathrm{PGL}(2,11)$ conjugacy classes of $\mathrm{Aut}(M_{12})$

Norton and Wilson - Maximal subgroups of the Harada–Norton group seems to imply that $\mathrm {HN} \rtimes C_2$ works, along with its overgroups $\mathbb{B}$ and $\mathbb{M}$.

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