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      Questions tagged [finite-groups]

      Questions on group theory which concern finite groups.

      6
      votes
      1answer
      201 views

      Open problems concerning all the finite groups

      What are the open problems concerning all the finite groups? The references will be appreciated. Here are two examples: Aschbacher-Guralnick conjecture (AG1984 p.447): the number of conjugacy ...
      2
      votes
      0answers
      54 views

      A question on verifying the mixing time of finite groups such as the Rubik's Cube Group

      I'm interested in some questions about the computational complexity of bounding the mixing time of random walks on Cayley-graphs of finite groups like that of the Rubik's Cube Group $G$. Determining ...
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      votes
      0answers
      49 views

      Relations of minimal number of generators

      What is the command in GAP to find the all relations of minimal generators of a finite $p$-group $G$?
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      votes
      0answers
      93 views

      On covers of groups by cosets

      Suppose that ${\cal A}=\{a_sG_s\}_{s=1}^k$ is a cover of a group $G$ by (finitely many) left cosets with $a_tG_t$ irredundant (where $1\le t\le k$). Then the index $[G:G_t]$ is known to be finite. In ...
      1
      vote
      0answers
      56 views

      Unit-product sets in finite decomposable sets in groups

      A non-empty subset $D$ of a group is called decomposable if each element $x\in D$ can be written as the product $x=yz$ for some $y,z\in D$. Problem. Let $D$ be a finite decomposable subset of a ...
      4
      votes
      1answer
      105 views

      Diameter for permutations of bounded support

      Let $S\subset \textrm{Sym}(n)$ be a set of permutations each of which is of bounded support, that is, each $\sigma\in S$ moves $O(1)$ elements of $\{1,2,\dotsc,n\}$. Let $\Gamma$ be the graph whose ...
      4
      votes
      1answer
      56 views

      Example of primitive permutation group with a regular suborbit and a non-faithful suborbit

      I would like some examples of groups $G$ satisfying all of the following criteria: $G<Sym(n)$, the symmetric group on $n$ letters, and $G$ is primitive. $G$ has a regular suborbit, i.e. if $M$ is ...
      4
      votes
      1answer
      172 views

      Is $PSL(2,13)$ a chief factor of the automorphism group of a $\{2,3\}$-group?

      Does there exists a group $H$ of order $2^7\cdot 3^4$, such that $\mathrm{PSL}(2, 13)$ is a chief factor of $\mathrm{Aut}(H)$?
      19
      votes
      2answers
      464 views

      Does the symmetric group $S_{10}$ factor as a knit product of symmetric subgroups $S_6$ and $S_7$?

      By knit product (alias: Zappa-Szép product), I mean a product $AB$ of subgroups for which $A\cap B=1$. In particular, note that neither subgroup is required to be normal, thus making this a ...
      4
      votes
      2answers
      168 views

      Which dimensions exist for irreducible quaternionic-type real representations of finite groups?

      I'm writing a software package to decompose group representations, and am struggling to find good examples of quaternionic-type representations of dimension > 4. Reading MathOverflow, I found that ...
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      votes
      0answers
      58 views

      Isomorphism of finite groups and cycle graphs

      Let $G$ and $H$ be finite groups and suppose they do have the same cycle graph. Is it possible to argue that this implies $G$ and $H$ are isomorphic? If yes, why? If not, is there an explicit ...
      5
      votes
      0answers
      150 views

      Breuer-Guralnick-Kantor conjecture and infinite 3/2-generated groups

      A group $G$ is called $\frac{3}{2}$-generated if every non-trivial element is contained in a generating pair, i.e. $$\forall g \in G \setminus \{e \}, \ \exists g' \in G \text{ such that } \langle g,g'...
      7
      votes
      1answer
      123 views

      Cycle types of permutations from affine group

      Let $V$ be a vector space of dimension $n$ over the field $F=\mathrm{GF}(2)$. We identify $V$ with the set of columns of length $n$ over $F$. Let $G = \mathrm{AGL}(V)$ be a group of affine ...
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      votes
      0answers
      119 views

      Growth functions of finite group - computation, typical behaviour, surveys?

      Looking on the growth function for Rubik's group and symmetric group, one sees rather different behaviour: Rubik's growth in LOG scale (see MO322877): S_n n=9 growth and nice fit by normal ...
      3
      votes
      0answers
      162 views

      Frobenius formula

      I know two formulas by the name of Frobenius. The first one computes the number $$\mathcal{N}(G;C_1,\dotsc,C_k):=|\{(c_1,\dotsc,c_k)\in C_1 \times \cdots \times C_k\:|\:c_1\cdots c_k=1\}|,$$ where $...

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