# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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### Are almost all permutation configurations from $S_n$ covered by small subsets subgroups of $S_n$?

Given integer $m\in[1,n]$ fix a set $\mathcal T$ of permutations in $S_n$. Then there are subgroups $G_1,\dots,G_m$ of $S_n$ so that $\mathcal T$ is covered by cosets of $G_1,\dots,G_m$.
Do we ...

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### A simple example of Galois extension with an arbitrary finite group [closed]

Does there exist a simple way to construct the Galois extension of some field with an arbitrary finite group?

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### Is every finite quantum group a quantum symmetry group?

This post is basically a quantum extension of Is every finite group a group of “symmetries”?
Here finite quantum group means finite dimensional Hopf ${\rm C}^{\star}$-algebra.
Frucht's theorem ...

**6**

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**1**answer

426 views

### Riemann Hypothesis, Primes and Groups

Let $G$ be a finite group $S\subset G$ a generating set $|g|=$ word length with respect to $S$. Set
$$ \sigma(G) = \sum_{H \le G} [G:H]$$
Let $\rho$ be the regular representation and set $A_G := \...

**16**

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474 views

### How is this group theoretic construct called?

Let $G$ be a finite group, $S\subset G$ a generating set, $|g| = |g|_S = $ word length with respect to $S$. Define the "defect" of $g,h$ to be
$$\psi(g,h) = |g|+|h|-|gh|$$
Then $\psi:G\times G \...

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**2**answers

131 views

### Generators for permutation groups

Consider (e.g.) the full permutation group $G=S_6$. A valid set of generators and equations for $G$ is $r^6=m^2=(rm)^5=1$. I say this system has width $3$ (because there are $3$ equations), length $10$...

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116 views

### Greatest Common Divisor of two specified sequences of numbers (search for equality)

I consider two sequences of numbers $A=\{a_1,...,a_n\}$ and $B=\{k-a_1,...,k-a_n\}$, where $a_1 \le a_2 \le ... \le a_n \le k$.
I am looking for such conditions under which: $gcd(a_1,...,a_n) = gcd(k-...

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32 views

### Crossed product with the same induced action in blocks of finite groups

Let $k$ be an algebraically closed field with characteristic $p$, let $G$ be a finite group such that $p$ divides $|G|$, let $N$ be a normal subgroup of $G$ with $p'$ index. Let $B$ be a block of $kG$,...

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217 views

### A question concerning some group action

Let $G$ be a finite group. Consider the set
$$X = \bigcup_{H \le G} G/H$$
which is a disjoint union of left cosets of subgroups $H$ of $G$.
Then $G$ acts on $X$ by left multiplication, and the number $...

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100 views

### Graeco-Latin squares and outer-automorphisms

It is well known that $n=6$ is the only number greater than two in which there is no Graeco-Latin square of order $n$. It is also well known that $n=6$ is the only number greater than two in which ...

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613 views

### What properties characterize the function $L(x) = x+\exp(x) \log(x)$?

As might be known, the function $L(x) = x+\exp(x)\log(x)$ plays a prominent role in the Lagarias formulation of the Riemann hypothesis:
$$\sigma(n) \le H_n + \exp(H_n) \log(H_n)$$
My question is, ...

**10**

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**1**answer

504 views

### A group theoretic interpretation of Lagarias inequality

Let $G$ be a finite group, $S \subset G$ a generating set. Set $\sigma(G):=\sum_{U \subset G} |U| $, where the sum runs over all subgroups $U$ of $G$. Set $H_G := \sum_{g \in G} \frac{1}{|g|+1}$, ...

**3**

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186 views

### Connected permutation groups and wreath product

Let $G$ and $H$ be subgroups of the symmetric groups $\mathfrak S_m$ and $\mathfrak S_n$. Assume that $n>1$ and that $H$ is a 'connected' permutation group, that is, there is no non-trivial $H$-...

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176 views

### What's concrete model for Coxeter complexes?

We know for every Coxeter system $(W,S)$ there is a Coxeter complex associated by its cosets of parabolic subgroups. In Wachs's note Poset Topology p.12-13 she mentioned for the Coxeter complex of ...

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148 views

### What are the points about representation of groups? [closed]

For a fixed (let say finite-, or Lie-, to respect the historical motivations) group, why does the study of all its linear representations over a fixed field, leads to some knowledge about its ...