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        Questions tagged [finite-groups]

        Questions on group theory which concern finite groups.

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        5
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        0answers
        116 views

        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 ...
        2
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        0answers
        98 views

        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 \...
        5
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        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 ...
        3
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        0answers
        111 views

        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
        votes
        1answer
        310 views

        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 ...
        2
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        0answers
        185 views

        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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        2answers
        466 views

        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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        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 ...
        1
        vote
        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 ...
        -3
        votes
        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 ...
        8
        votes
        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 (...
        0
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        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|...
        4
        votes
        0answers
        120 views

        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
        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 $486$ ...
        2
        votes
        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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