# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

1,571
questions

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### Explicit description of the smallest class of groups, that contains all finite simple groups and is closed under semidirect products

Suppose $\Pi$ is the smallest class of groups satisfying the following conditions:
All finite simple groups lie in $\Pi$
If $G \cong H \rtimes K$ and both $H$ and $K$ are in $\Pi$, then $G$ is also ...

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### On group varieties and numbers

Suppose $\mathfrak{U}$ is a group variety. Let’s define $N_{\mathfrak{U}} \subset \mathbb{N}$ as a such set of numbers, that for any finite group $G$, $|G| \in N_{\mathfrak{U}}$ implies $G \in \...

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### Are finite groups of exponent $d$ rare for $d \neq 4$?

Is there a way to prove, that $\lim_{n \to \infty} \frac{\text{the number of all groups of exponent }d \text{ and order less than }n}{\text{the number of all groups of order less than } n} = 0$ for $d ...

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### A question on a result of Imre Ruzsa concerning sum-sets

Th main result of this preprint of Imre Ruzsa implies the following
Corollary (Ruzsa): For every $r\in\mathbb N$ there exists a real number $\alpha<1$ and a positive integer $m$ such that for ...

**4**

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**1**answer

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### Is there some sort of classification of finite groups that force solvability?

Suppose $G$ is a finite group. We will say, that it force solvability if any finite group $H$, such that $G$ is isomorphic to its maximal proper subgroup, is solvable. Does there exist some sort of ...

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### Abstract of talk by Wielandt required

I am searching for Abstracts of short communications of the International Congress of Mathematicians, 1962. In particular, the abstract of Wielandt's talk "Bedingungen für die Konjugiertheit von ...

**10**

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### A criterion for finite abelian group to embed into a symmetric group

Let $G$ be a finite abelian group. Write $G\approx \mathbb{Z}/p_1^{i_1}\mathbb{Z}\times\dots \mathbb{Z}/p_m^{i_m}\mathbb{Z}$, with $m\ge 0$, $p_1,\dots,p_m$ primes (not necessarily distinct) and $i_k\...

**8**

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### Is there a way to prove, that $2$-generated groups are rare among finite groups?

Is there a way to prove, that $\lim_{n \to \infty} \frac{\text{the number of all } 2 \text{-generated groups of order less than }n}{\text{the number of all groups of order less than } n} = 0$?
This ...

**1**

vote

**1**answer

207 views

### What finite simple groups appear as factors of surface fundamental groups?

Let $\Sigma_g$ be the a closed orientable surface of genus $g$.
My somewhat naive question: what is known about simple finite factors of $\pi_1(\Sigma_g)?$ In particular, I know that the composition ...

**-3**

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**1**answer

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### What do you call continous transformations that preserve the finite group structure?

A number of years ago I studied a preon model (Journal of Mathematical Physics 38:3414-3426, 1997) in which the preons interacted like group elements. I thought it curious that you could sometimes ...

**8**

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578 views

### Groups without factorization

A group G is said to have a factorization if there exist proper subgroups $A$ and $B$ such that $G = AB = \{ ab \ | \ a \in A, b \in B \}$.
The paper Factorisations of sporadic simple groups (...

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**1**answer

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### Confusion on translating k-fold transitivity of groups from Endliche Gruppen by Huppert

The definition 1.7 from Endliche Gruppen, B.Huppert, vol-I, Chap.II, Pg.148 is as follows: Die Permutationsgruppe $\mathfrak G$ auf der Ziffernmenge $\Omega$ hei?t $k$-fach transitiv $(k \leq |\Omega|...

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### A group-theoretical analogous of Temperley-Lieb-Jones subfactor planar algebras

The Temperley-Lieb-Jones subfactor planar algebra $\mathcal{TLJ}_{\delta}$ admits the following properties:
maximal,
it exists for every possible index, i.e. $\delta^2 \in \{4cos^2(\pi/n) \ | \ n \...

**5**

votes

**1**answer

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### Are there infinitely many insipid numbers?

A number $n$ is called insipid if the groups having a core-free maximal subgroup of index $n$ are exactly $A_n$ and $S_n$. There is an OEIS enter for these numbers: A102842. There are exactly $486$ ...

**2**

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**1**answer

295 views

### The sporadic numbers

Let call $n$ a sporadic number if the set of groups $G \neq A_n,S_n$ having a core-free maximal subgroup of index $n$ is non-empty and contains only sporadic simple groups.
By GAP, the set of all the ...