# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

1,498 questions

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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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### 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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91 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 ...

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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**

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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 ...

**4**

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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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457 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**

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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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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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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 ...