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        Questions tagged [finite-groups]

        Questions on group theory which concern finite groups.

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        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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        93 views

        Fourier transforms on finite abelian groups [closed]

        Is the Fourier transform of a product of finite abelian groups equivalent to the product of the transforms, i.e. $$ \hat{f}\big(\prod_i G_i\big) = \prod_i \hat{f}(G_i),$$ where the $G_i$ are finite ...
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        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 ...
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        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 ...
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        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 ...
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        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 ...
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        1answer
        169 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 $\mathrm{Aut}(H)$?
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        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 ...
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        167 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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        56 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 ...
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        148 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'...
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        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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        117 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 ...
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        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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        1answer
        233 views

        What is the minimum $k$ such that $A^k \equiv I$ mod p for invertible matrices?

        Let $F$ be a finite field of order $p$, where $p$ is prime. For any $n\times n$ matrix $A$ that is invertible over $F$, then there would appear to exist integers $k$ such that $A^{k} = I$. My question ...

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