# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

1,499 questions

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### 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**

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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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### 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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### 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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**1**answer

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**

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**1**answer

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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**1**answer

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**

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**2**answers

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**

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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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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 ...

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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**

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**1**answer

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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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**

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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 $...