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# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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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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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 $G \in \... 0answers 173 views ### Are finite groups of exponent$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 ...
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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 ...
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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\... 1answer 296 views ### 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 ... 1answer 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 ... 1answer 156 views ### 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 ... 2answers 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 (... 1answer 74 views ### 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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The Temperley-Lieb-Jones subfactor planar algebra $\mathcal{TLJ}_{\delta}$ admits the following properties: maximal, it exists for every possible index, i.e. $\delta^2 \in \{4cos^2(\pi/n) \ | \ n \... 1answer 2k views ### Are there infinitely many insipid numbers? A number$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$486$... 1answer 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 ...

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