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4 votes
0 answers
180 views

Subgroups that conjugate-cover the ambient group

Let $G$ be a finite group, and suppose that a set of proper subgroups $H_1,\dotsc,H_n$ satisfy $G=\bigcup_{g\in G}\bigcup_{i=1}^nH_i^g$, where $H_i^g$ is the conjugate of $H_i$ by $g$. In this case, ...
Nicolas Banks's user avatar
1 vote
0 answers
85 views

inverse Galois problem on cyclic groups

It is known that the splitting field of $x^{p^n}-x$ over $\mathbb{F}_p$ is $\mathbf{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)\cong\mathbb{Z}/n\mathbb{Z}$ and the splitting field of $\Phi_n(x)$ over $\mathbb{...
user537897's user avatar
7 votes
1 answer
282 views

Galois groups of truncated $\cosh(x)$ Taylor polynomials and related results?

(Part of this question was written with ChatGPT because english is not my native language). I am currently translating my diploma thesis from 2010 in english and thought to think about the topic again:...
mathoverflowUser's user avatar
5 votes
1 answer
358 views

The number of polynomials on a finite group, II

This question is follow up of this MO-post. First let us recall the necessary definitions. A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\mathbb N$ and elements $a_0,...
Taras Banakh's user avatar
  • 41.8k
2 votes
0 answers
46 views

Finite groups whose polynomials share two common properties with polynomials on commutative groups

This question is motivated by (some available information on) this MO-problem on the largest possible degree of a polynomial on a finite group and this MO-problem on the degree of the constant ...
Taras Banakh's user avatar
  • 41.8k
9 votes
1 answer
482 views

The degree of a constant polynomial on a finite group

A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\mathbb N=\{1,2,\dots\}$ and elements $a_0,a_1,\dots,a_n\in X$ such that $f(x)=a_0xa_1x\cdots xa_n$ for all $x\in X$. ...
Taras Banakh's user avatar
  • 41.8k
4 votes
1 answer
264 views

A question on a possible cyclic sieving phenomenon?

(This is an old MSE question from me, which did not get any answer, and when looking back seems interesting to post it here:) Let $G$ be a finite group. Consider the set $X_G:=\cup_{H\le G} G/H$, ...
mathoverflowUser's user avatar
13 votes
0 answers
247 views

Galois groups of special polynomials

This question is motivated by long experiments with GAP. Call a monic polynomial with integer coefficients special in case it is irreducible and has only coefficients $-1$, $0$ or $1$. Let $n \geq 5$....
Mare's user avatar
  • 26.5k
1 vote
1 answer
874 views

About Frobenius Determinant Theorem

Finite group $G=\{x_1,x_2,...x_n\}$. Consider $G$'s multiplication table to be an $n\times n$ matrix $A$. Set $x_i=1$, $x_j=0$ ($j≠i$), $1≤i≤n$, then we get $n$ permutation matrices $S_i$ ($1≤i≤n$) s....
user119016's user avatar
28 votes
0 answers
676 views

Mathieu group $M_{23}$ as an algebraic group via additive polynomials

An elegant description of the Mathieu group $M_{23}$ is the following: Let $C$ be the multiplicative subgroup of order $23$ in the field $F=\mathbb F_{2^{11}}$ with $2^{11}$ elements. Then $M_{23}$ is ...
Peter Mueller's user avatar
2 votes
0 answers
146 views

A question about an irreducible polynomial in the ideal of syzygies

Let $G$ be a finite group with $n$ elements and let $\mathbb{Q}[x_1,\cdots,x_n]^G = \mathbb{Q}[g_1,\cdots,g_m]$, where $G$ acts through the regular representation. Then there exist polynomials $s_j \...
user avatar
2 votes
1 answer
240 views

A question about the invariants of a finite group

Let $G$ be a finite group with $n$ elements and let $\mathbb{Q}[x_1,\cdots,x_n]^G = \mathbb{Q}[g_1,\cdots,g_m]$. Then there exist polynomials $s_j \in \mathbb{Q}[y_1,\cdots,y_m]$ for $j=1,\cdots,n$ ...
user avatar
15 votes
1 answer
825 views

Weyl Groups as Galois groups

I am looking for explicit examples (for all positive integers $n \ge 5$) of degree $2n$ even polynomials $f(x)=h(x^2)$ over the field $\mathbb{Q}$ of rational numbers such that the Galois groups of $...
Yuri Zarhin's user avatar
  • 5,050
12 votes
2 answers
660 views

On shifted symmetric power sums

The functions $p^*_k(x)=\sum_{i=1}^N ((x_i-i)^k-(-i)^k)$ are analogues of power sum symmetric functions, called shifted symmetric by Okounkov and Olshanski. Define $p^*_{(k_1,k_2,...)}=p^*_{k_1}p^*_{...
Marcel's user avatar
  • 2,552
13 votes
2 answers
2k views

Galois group of a product of polynomials

How can I compute the Galois group of the polynomial $fg\in K[x]$ assuming that I know the Galois groups of $f\in K[x]$ and $g\in K[x]$? Let's suppose for simplicity that the field $K$ is perfect.
roger123's user avatar
  • 2,782