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      Questions tagged [finite-groups]

      Questions on group theory which concern finite groups.

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      3answers
      156 views

      Subgroup generated by a subgroup and a conjugate of it

      Let $H\leq G$ be groups, and $a\in G$ so that $\langle H,a\rangle=G$. Does it follows that $\langle H\cup aHa^{-1}\rangle$ is a normal subgroup of $G$? My hope is that this is true, and my guess is ...
      5
      votes
      0answers
      120 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
      votes
      0answers
      100 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
      votes
      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
      votes
      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
      votes
      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
      votes
      2answers
      468 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
      votes
      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
      157 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
      votes
      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 $486$ ...

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