Questions on group theory which concern finite groups.

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**1**answer

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### Center of an irreducible representation over $\mathbb{Q}$

Let $G$ be a finite group, $\rho\colon G \rightarrow \mathrm{GL}_n(\mathbb{Q})$ its irreducible representation, and $D$ the division algebra of $G$-endomorphisms of $\mathbb{Q}^n$. The division ...

**16**

votes

**4**answers

712 views

### What algebraic structures are related to the McGee graph?

Recall that an $(n,g)$-graph is a simple graph where each node has $n$ neighbors and the shortest cycle has length $g$, while an $(n,g)$-cage is $(n,g)$-graph with the minimum number of nodes.
The ...

**30**

votes

**3**answers

1k views

### How much of the ATLAS of finite groups is independently checked and/or computer verified?

In a recent talk Serre made some comments about proofs that rely on the classification of finite simple groups (CFSG) and on the ATLAS of Finite Groups. Namely, he said that a proof that relied on the ...

**2**

votes

**1**answer

188 views

### Finite sequences which happen to be the sequence of orders of elements of a simple group

Let $n\in \Bbb N$. Any finite group $G$ with $|G|=n$ has a solution for the Diophantine equation
$$\sum_{d|n}x_d\phi(d)=n~~~~~~~~~~~~~~~~~~~~~~~(1)$$
where $\phi$ is the Euler's totient function, $d$ ...

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votes

**1**answer

82 views

### On Sylow subgroup of a finite group [closed]

Let $p\mid n$, then by $n_p$ we mean the $p$-part of $n$, i.e. $n_p = p^k$ if $p^k\mid n$ but $p^{k+1}\nmid n$. Let $G$ be a finite group, $M\leq G$ and $P\in Syl_p(G)$. Is It true that $|M\cap ...

**3**

votes

**1**answer

123 views

### Classification of finite abelian hypergroups and table algebras

Update: Originally, I formulated this question for finite abelian hypergroups, but in a discussion with Geoff Robinson below I realized that the abelian hypergroups defined below are equivalent to ...

**1**

vote

**1**answer

259 views

### Classification of finite group schemes over a field

What is known about the classification of finite group schemes over a field? By a finite group scheme I mean $Spec A$ where $A$ is a finite-dimensional algebra over a field.
Is there a full ...

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votes

**2**answers

353 views

### How simple does a $\mathbb{Q}$-simple group remain after base change to $\mathbb{Q}_{\ell}$?

Of course the general answer to the question in the title is: not very simple.
I could not think of a better title, so let me explain my question in more detail.
I have a number field $E/\mathbb{Q}$, ...

**0**

votes

**0**answers

84 views

### When is a group generated by three involutions two of which commute a Coxeter group?

Let's say the group is generated by three involutions $a$, $b$, and $c$ such that $ord(ab)= 2$, $ord(bc)=3$, and $ord(ac)=m$. Under which conditions is it isomorphic to the rank 3 Coxeter group $(2, ...

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**4**answers

186 views

### Are the orders of the generators of a group and the product of pairs of thereof enough for this group to be isomorphic to a Coxeter group?

Let's say we have $n$ generators $x_1, x_2, \cdots, x_n$ along with the following facts concerning their orders:
\begin{eqnarray*}
ord(x_i) &=& 2 \text{ for } i = 1, 2, \cdots, n \\
ord(x_i ...

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votes

**0**answers

178 views

### On shifted symmetric power sums

The functions $p^*_k(x)=\sum_{i=1}^N ((x_i-i)^k-(-i)^k)$ are analogues of power sum symmetric functions, called shifted symmetric by Okounkov and Olshanski. Define ...

**1**

vote

**1**answer

92 views

### embedding of finite groups into product

Our situation is following. Assume that we have free product $\star_{i<n} G_i$ each $G_i$ finite group and assume that we have normal subgroup $K$ such that composition of canonical embedding and ...

**2**

votes

**1**answer

113 views

### What is a good program for matrix groups computations?

I need a computer program, to help me with some very basic group computations.
Specifically, I want to know if some group generated by a few small matrices over a finite field is solvable.
Is there a ...

**4**

votes

**0**answers

201 views

### Non averaging sequences in finite groups

Let us say that a non averaging sequence in a group $G$ is a sequence $x_1, \dots, x_n$ such that
$$x_i^2 \neq x_j x_k$$
for any indices $i, j, k$ such that two at least are distinct. My question is: ...

**0**

votes

**0**answers

57 views

### Commutator subgroup of rotational symmetries of the hypercube

I would like to know which is the commutator subgroup of the group of rotational isometries of the $n$-dimensional cube. The group i am talking about is the subgroup of $\{ \pm 1 \} \wr S_n$ ...

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**0**answers

109 views

### Classification of Automorphism set of a Regular graph

Let $A$ be the adjacency matrix of an $r$-regular graph $G$ with $n$ vertices (Not complete or cycle graph) . Also, let $Aut(G)$ be the set of all its automorphisms (i.e. set of permutation ...

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votes

**2**answers

436 views

### What is the permutation group generated by those three given morphisms of the affine space $\mathbb{F}_q^3$?

Let $\mathbb{A}^3 = \mathbb{F}_q^3$. Consider the following three functions $\mathbb{A}^3\to\mathbb{A}^3$:
\begin{eqnarray*}
h: (x, y, z) &\mapsto& (x, y, xy - z) \\
u: (x, y, z) ...

**1**

vote

**1**answer

244 views

### Generalization of a theorem of Steinberg

Steinberg has a beautiful theorem counting $F$-stable maximal tori in a reductive group. Here's the version of the result that you can find as Theorem 3.4.1 of Carter's Finite groups of Lie type:
...

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votes

**1**answer

687 views

### Which finite groups have no irreducible representations other than characters?

A classical result states that all the irreducible representations of a finite group over $\mathbb{C}$ are characters if and only if $G$ is abelian. I would like to know what happens if we consider a ...

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vote

**1**answer

129 views

### Involutive automorphisms of a finite abelian p-group

First, let $A$ be a finitely generated free abelian group, and $s$ an automorphism of order $2$ of $A$. Set $G=\{1,s\}$. Then we know that $A$ is a sum of indecomposable $G$-lattices $A_i$, where ...

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votes

**4**answers

445 views

### SO$(4)$ (& SO$(n)$) characterization?

I believe it is the case that
any finite subgroup of SO$(3)$
(the $3 \times 3$ orthogonal matrices of determinant $1$)
is either a cyclic group $C_n$,
or a dihedral group $D_n$, or one of the groups ...

**2**

votes

**1**answer

127 views

### Counting elements with certain word length in abelian groups

Given a (finite) abelian group $G = \langle S \mid R \rangle$, has the problem of counting the number of elements which can be expressed as a word (in $S$) of length $\leq k$ been studied? If so, ...

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votes

**1**answer

162 views

### Is there a way to find an efficient set of relations for presenting the subgroup generated by two matrices in $SL(2, q)$?

Given two elements $a, b \in SL(2, \mathbb{F}_q)$, is there a way to find an efficient presentation $$\langle x, y \mid \text{relations}\rangle$$ of the subgroup $\langle a, b \rangle$?
My intention ...

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votes

**2**answers

130 views

### What are the 2-generated subgroups of the special linear group $SL(2, q)$ over a finite field?

What is the subgroup structure of the subgroups $\langle a, b\rangle$ where $a, b \in SL(2, q)$?

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votes

**1**answer

232 views

### Is every nonabelian finite simple group a quotient of a triangle group $(a,b,c)$ with $a,b,c$ coprime?

Here by triangle group $(a,b,c)$ I mean the group with presentation
$$\langle x,y \;|\; x^a = y^b = (xy)^c = 1\rangle$$
In other words, for every finite simple nonabelian group $G$, do there exist ...

**4**

votes

**1**answer

523 views

### Is there a structure theorem or group law for finite groups generated by two elements?

Say that $a, b \in G$ are two elements of a finite group $G$. Is there a structure theorem for the structure of $\langle a,b\rangle$? Is there a way to derive group laws for the group operation in the ...

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votes

**1**answer

170 views

### A representation of a finite group where every nonzero vector has a trivial stabilizer [duplicate]

What are the finite groups which admit a non-zero representation in char 0 where every non-zero vector has stabilizer equal to $\left<1\right>$? Cyclic groups of prime order is one obvious ...

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votes

**0**answers

196 views

### A class 3 group of order 243

Let G be a group of order $243=3^5$. We denote by $(G_i)$ its lower central series and assume that $G$ has class $3$ and that $|G:G_2|=|G_3|=9$. We assume moreover that the cubing map factors as a ...

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votes

**1**answer

3k views

### Why can't a nonabelian group be 75% abelian?

This question asks for intuition, not a proof.
An earlier question,
Measures of non-abelian-ness
was thoroughly answered by Arturo Magidin.
A paper by Gustafson1
proves that, for a nonabelian group,
...

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votes

**0**answers

34 views

### Is there a non-solvable integral fusion category of square-free dimension?

A finite group of square-free order is solvable (see here).
You can find the definition for a solvable fusion category in this paper.
Question: Is there a non-solvable integral fusion category of ...

**2**

votes

**0**answers

199 views

### Does knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ sometimes offer a significant advantage in finding $r$?

Is there a cyclic group $\mathcal G$ with generator $g$ for which the discrete log problem is assumed to be hard, but knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ for random $r$ makes finding $r$ easy?
...

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votes

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176 views

### Isotropic subspaces in a symplectic vectorspace over $GF(q)$

Let $V$ be a symplectic vectorspace of dimension $2n$, and $r\mid n$. Is this statement true?"There is an isotropic spread of $r$ dimensional subspaces in $V$". By an isotropis subspace I mean a ...

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vote

**2**answers

267 views

### Subgroup of Projective general linear group on complete discrete valuation ring

Let $R$ be a complete dvr and $k$ its residue field of positive characteristic.
Let $H$ be a finite subgroup of $PGL_2(k)$ such that the order of $H$ is prime with $char(k)$.
Is there some ...

**2**

votes

**1**answer

149 views

### Asymptotic of min(#generators times diameter), for a Cayley graph of Sn

This post is a sequel of Diameter of symmetric group.
Let $\Sigma$ a generating subset of $S_n$, $\Gamma(S_n, \Sigma)$ the Cayley graph and $d_{\Sigma}$ the diameter of $\Gamma(S_n, \Sigma)$.
Let ...

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votes

**2**answers

173 views

### Permutation covering of a $G$-lattice

Let $G$ be a finite group.
By a $G$-lattice we mean a finitely generated free abelian group $L$ with an action of $G$.
We say that $L$ is a permutation $G$-lattice if $L$ has a ${{\mathbf{Z}}}$-basis ...

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votes

**0**answers

260 views

### On Thompson conjecture

Let $N(G)$ be the set of conjugacy class sizes of finite group $G$. Let $G$ be a finite group for which $N(G)=N(A_n)$, where $A_n$ is the alternating group of degree $n$.
Let $p$ and $q$ be distinct ...

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502 views

### Next steps on formal proof of classification of finite simple groups

While people are steaming ahead on finessing the proof of the classification of finite simple groups (CFSG), we have a formal proof in Coq of one of the first major components: the Feit-Thompson ...

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128 views

### Representation of finite group

Let $\Gamma$ be a finite index subgroup of $SL_2(\mathbb{Z})$. Let $\pi$ be a natural covering from $\Gamma\backslash\mathbb{H}$ to $SL_2(\mathbb{Z})\backslash\mathbb{H}$. Denote by $\text{Deck}(\pi)$ ...

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157 views

### Finite groups which have elements $g$ of order $pq$ such that the sizes of the conjugacy classes of $g^p$, of $g^q$ and of $g^{q-p}$ coincide [closed]

Let $G$ be a finite group and let $g \in G$ be an element of order $pq$, where $p < q$ are
prime numbers. Denote by $g^G$ the conjugacy class of $g$ in $G$. Under which conditions
does the ...

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vote

**0**answers

165 views

### Are the finite groups inclusions, almost all relatively cyclic?

Definition: An inclusion of finite groups $(A \subset B)$ is relatively cyclic if $\exists b \in B$ such that $\langle A,b \rangle = B$.
Definition: Two inclusions of finite groups are equivalent, ...

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vote

**1**answer

130 views

### Groups with many vanishing elements

It is well-known that every non-linear character $\chi$ of a finite group $G$ vanishes on some elements of $G\setminus Z(\chi)$. The question is
What can be said about a finite group $G$ for which ...

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votes

**2**answers

496 views

### Origin of group theory problem (bound on number of Sylow subgroups)

This problem (prove that the number of Sylow subgroups of a finite group $G$ is bounded by $\frac{2}{3}|G|$) posted on MSE proved rather difficult to solve. The OP has been silent about where the ...

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votes

**1**answer

196 views

### Finite quotients of an infinite product of finite groups

Let $G$ be a finite group.
Consider the direct product $\Gamma=\prod_{i=1}^{\infty}G$ of (countably) infinitely many copies of $G$. For every finite set of numbers $\{i_1,\ldots,i_n\}$ we have the ...

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votes

**2**answers

584 views

### Is there a big solvable subgroup in every finite group?

Definition: Let $G$ be a group, and let $H \leq G$ be a subgroup. We say that $H$ is big in $G$ if for every intermediate subgroup $H \leq L \leq G$ there exists some $x \in L$ such that $\langle H ...

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**0**answers

111 views

### Subgroups of powers of the alternating group on 5 elements

Definition: Let $G$ be a group, and let $H \leq G$ be a subgroup. We say that $H$ is big in $G$ if for every intermediate subgroup $H \leq L \leq G$ there exists some $x \in L$ such that $\langle H ...

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65 views

### When does a finite group have a lower-dimensional representation than one of its quotients?

The minimal dimension of a nontrivial representation of a central extension of a group $G$ can be smaller than the minimal dimension of a nontrivial representation of $G$. What about non-central ...

**17**

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**2**answers

582 views

### divisors of $p^4+1$ of the form $kp+1$

In group theory the number of Sylow $p$-subgroups of a finite group $G$, is of the form $kp+1$.
So it is interesting to discuss about the divisors of this form. As I checked it seems that for an odd ...

**4**

votes

**1**answer

166 views

### The groups with symmetric subgroups lattice

Let $G$ be a group and $\mathfrak{L}(G)$ be set of all subgroups of $G$. Clearly, $\mathfrak{L} (G)$ is a lattice.
If we know that $\mathfrak{L} (G)$ is symmetric then what can be said about the ...

**1**

vote

**1**answer

189 views

### Is a prime index inclusion of finite groups, separating?

Let $(H \subset G)$ be an inclusion of finite groups.
Let $\{ g_i \ \vert \ i \in I=[1, \dots ,n] \}$ a subset of $G$ of double coset representatives, i.e. $$G = \coprod_{i \in I} Hg_iH$$
On the ...

**4**

votes

**0**answers

60 views

### Minimal rank of a permutation resolution of a $G$-lattice

Let $G$ be a finite group.
By a $G$-lattice I mean a finitely generated free abelian group $L$ with an action of $G$.
One says that $L$ is a permutation lattice if $L$ has a $\mathbb{Z}$-basis ...