# Tagged Questions

Questions about the branch of abstract algebra that deals with groups.

**18**

votes

**1**answer

624 views

### Have finite doubly transitive groups been classified?

I am trying to determine whether the literature contains a complete proof of the classification of finite 2-transitive groups. This is a fundamental result with important applications in many areas ...

**4**

votes

**1**answer

266 views

### Do irreducible characters form a closed set?

A character on a discrete group $\Gamma$ is a conjugation-invariant function $\tau$ which is of positive
type, and is normalized so that $\tau(e) = 1$, where $e$ is the identity element of $\Gamma$. A ...

**1**

vote

**0**answers

89 views

### When does a normal subgroup H of a group G have a complement in G? [closed]

When does all normal subgroups of a group have complement? This question is different from question When does a subgroup H of a group G have a complement in G?
Related to this question I ask is ...

**5**

votes

**1**answer

168 views

### CAT(0)-groups in dimension 2

Suppose I have a space $X$ which is connected, simply connected, CAT(0) of dimension 2 and a group $G$ which acts on $X$ freely, isometrically, properly discontinuously and cocompactly. What can be ...

**6**

votes

**3**answers

413 views

### Solvable irreducible subgroups of the $\mathbf{GL}_n$ of $\mathbf{F}_p$ ($p$ prime)

I have a finite-dimensional vector space $E$ over the finite prime field $\mathbf{F}_p$ and a solvable subgroup $G\subset\mathbf{GL}(E)$ for which $E$ is an irreducible representation. Do the groups $...

**6**

votes

**1**answer

169 views

### Real-valued character in Block with cyclic defect has at most two constituents modulo $p$

Let $G$ be a finite group and let $(K,R,k)$ be a $p$-modular system (large enough for $G$ etc.) and consider a block algebra $B \subseteq RG$ with cyclic defect group.
My question is about the ...

**6**

votes

**1**answer

463 views

### Non-vanishing of the Tate-Shafarevich kernel in group cohomology

Let $G$ be a finite group. Let $M$ be a finite $G$-module (a finite abelian group with an action of $G$).
We consider a special kind of $G$-modules; in particular, our $M$ is a finite dimensional ...

**5**

votes

**0**answers

101 views

### “Factorisation” in special linear groups over rings of integers

It is known that for any number field $F$ with infinitely many units (i.e. $F$ is not $\mathbb Q$ or an imaginary quadratic field) with ring of integers $O$ the special linear group $\mathrm{SL}_2(O)$ ...

**5**

votes

**1**answer

170 views

### Subgroup ranks of the symmetric group

It's well known that every subgroup $G$ of $S_n$ has a generating set of size at most $n-1$ and that this generating set can be found algorithmically (by Jerrum's filter)
I have heard many times a ...

**3**

votes

**2**answers

267 views

### Determining whether or not a subset of $S_n$ generates $S_n$

I have a certain family of subsets of $S_n$, and I'd like to know which subsets in this family generate $S_n$. What techniques exist for solving this type of problem?
Are there any known results on ...

**4**

votes

**0**answers

59 views

### Is there an interval of finite groups, at index n, with strictly more elements than the subgroup lattice of any group, of order n?

Let $G$ be a finite group and $\mathcal{L}(G)$ its subgroup lattice.
Let $s(n):= max\{|\mathcal{L}(G)| \text{ for } |G|=n \}$.
There is an OEIS page for the sequence $s(n)$: A018216
1, 2, 2, 5, 2, ...

**3**

votes

**0**answers

109 views

### Large almost abelian subgroups of permutation groups

Let $m<n$. Let $t_{m,n}$ be the largest cardinality of a subgroup $G\subseteq S_{n}$ such that if $A\subseteq\{1,...,n\}$ and $|A|=m$, then the subgroup
$\{f\in G:f|_{A}=\text{Id}_{A}\}$ of $G$ is ...

**2**

votes

**0**answers

79 views

### Isomorphism with fixed number of Permutations [closed]

Suppose, I have a fixed number of permutations for each sub-graph to determine isomorphism of whole graph. Is it possible to determine efficiently ?
For example, $G, H$ are isomorphic graphs. For ...

**7**

votes

**1**answer

242 views

### Is it known whether every symmetric pair of finite groups of Lie type is a Gelfand pair?

A pair of groups $(G,H)$ is called a symmetric pair if $H$ is the group of fixed points of an involutive automorphism of $G$, for example $(GL(2n,\mathbb{F}_q),Sp(2n,\mathbb{F_q}))$ is a symmetric ...

**0**

votes

**1**answer

85 views

### Relation between the Frattini Property and Pronormal subgroups of Solvable groups

A subgroup $H$ of $G$ is said to satisfy the Frattini Property if for any subgroup $K$ and $L$ such that $H\leq K \unlhd L$ implies that $L \leq N_L(H)K$
A subgroup is $H$ is pronormal in $G$ if for ...

**1**

vote

**2**answers

230 views

### A good upper-bound for the cardinal of an interval of finite groups

This post is a relative version of General bound for the number of subgroups of a finite group
Let $[H,G]$ be a interval of finite groups with $|G:H| = n$.
Question: What is a good upper-bound of $|[...

**0**

votes

**0**answers

76 views

### Generating set of Graph-Automorphism from Direct Product

Notation:
$H$ is the adjacency matrix of graph $\mathcal{H}$ .
$$H = \begin{bmatrix}
H_{(3)} & R_{(3, 2)} & R_{(3,1)} \\
R_{(3,2)} & H_{(2)} & R_{(2,1)} \\
R_{(3,1)} & R_{(2,1)}...

**31**

votes

**3**answers

2k views

### Is there a nice explanation for this curious fact about cyclic subgroups?

Here's something that I noticed that quite surprised me.
Let $G$ be a finite abelian group. Consider the following expression.
$$
\nu(G) = \sum_{\substack{H \leq G \\ H \text{ is cyclic}}} |H|
$$
It ...

**1**

vote

**0**answers

118 views

### Classification of finite subgroup of $PGSp_4(\mathbb{C})$

Is there a classification of the finite subgroups of $PGSp_4(\mathbb{C})$?

**4**

votes

**2**answers

229 views

### Historical reference request on Nilpotent groups

From Wikipedia:
"Abelian groups were named after Norwegian mathematician Niels Henrik Abel by Camille Jordan because Abel found that the commutativity of the group of a polynomial implies that the ...

**7**

votes

**1**answer

299 views

### Groupoid cardinality and Egyptian fraction representations of 1

It is well-known that any rational number can be represented using a sum of distinct Egyptian fractions (that is, rational fractions of the form $1/n$ with $n\in\mathbb{N}$). This may be proven by ...

**14**

votes

**1**answer

369 views

### Does the ring $R = \mathbb{Z}[X^{\pm1}]$ of Laurent polynomials over $\mathbb{Z}$ satisfy $SL_2(R) = E_2(R)$?

Let $R = \mathbb{Z}[X^{\pm1}]$ be the ring of Laurent polynomials on one indeterminate over $\mathbb{Z}$. Let $E_2(R)$ be the subgroup of $GL_2(R)$ generated by the matrices that differ from the ...

**4**

votes

**0**answers

169 views

### Is the class of commutative generalized Euclidean rings stable under quotient and localization?

Let $R$ be an associative ring with indentity and let $E_n(R)$ be the subgroup of $GL_n(R)$ generated by matrices obtained from the identity matrix by replacing an off-diagonal entry by some $r∈R$. ...

**1**

vote

**0**answers

59 views

### A non-surjective coboundary map induced by a central extension

Let $k$ be a number field and
$$ 1\to A \to B \to C \to 1$$ be a central extension of finite groups over $\mathcal{O}_k$ (the ring of integers of $k$), with $B$ non-commutative. Consider the induced ...

**4**

votes

**0**answers

160 views

### A metric on $Homeo([0,1])$

One can define a metric on the set $Homeo([0,1])$ by setting $dist(f,g) =$ measure of support of $f^{-1}g$, that is the measure of the set of points $x$ where $f(x)\ne g(x)$. Was this metric studied ...

**12**

votes

**2**answers

526 views

### 1-st cohomology of multiplicative group in a vector space

Let $\mathbb k$ be a field of characteristic $p$ and let $\mathbb k_n$ be a 1-dimensional representation of $\mathbb k^\times$, where the action is given by $t\circ v= t^n v$. Is it known what are the ...

**2**

votes

**2**answers

137 views

### Realization of irreducible $\mathfrak{S}_d$-modules and the representation theory of Lie algebra

Let $n$ be a positive integer. It is well-known that a method to realize irreducible $\mathfrak{S}_d$-modules is to construct the so-called Specht modules $S^{\mu}$ which are submodules in the so-...

**2**

votes

**0**answers

115 views

### Nonvanishing of the dual Euler totient on boolean intervals of finite groups

The rank $n$ boolean lattice $B_n$, is the subset lattice of $\{1,2, \dotsm n \}$.
Let $[H,G]$ be a boolean interval of finite groups. Its Euler totient is defined by $$\varphi(H,G):=\sum_{K \in ...

**4**

votes

**2**answers

225 views

### Does there exist a non-hyperelliptic Riemann surface with automorphism group $C_2\times A_4$?

Does there exist a non-hyperelliptic Riemann surface of genus 5 with automorphism group $C_2\times A_4$?

**3**

votes

**0**answers

77 views

### Correspondence between dual center and linear characters of finite reductive group

Let $(G,F)$ be a connected reductive group defined over $\mathbb{F}_q$ via the Frobenius $F$ and let $(G^*,F^*)$ be a group in duality with $(G,F)$ with respect to rational maximal tori $T \subseteq G$...

**1**

vote

**0**answers

97 views

### Presentation of hyperbolic groups [closed]

Is it true that all hyperbolic groups are finitely presented? If yes, what is the right reference for that?

**8**

votes

**2**answers

313 views

### Normal subgroups of Aut(M)

Let $S$ be the set of all finite permutations of $\mathbb{N}$, i.e. they fix all but a finite set, and $A\subset S$ the set of all even permutations.
Theorem The normal subgroups of $S_\infty$ are ...

**1**

vote

**1**answer

199 views

### Closed subgroups of $\mathrm{SO}(4)$

My question is quite simple : we know all closed subgroups of $\mathrm{SO}(3)$; is it also known what are the closed subgroups of $\mathrm{SO}(4)$?

**2**

votes

**0**answers

199 views

### Show that $SL_2(\mathbb{F}_p)$ is quasi-random

Terry Tao gives this oblique definition of quasirandom group in his notes 3
$G$ is quasi-random (of order $D$) if all non-trivial unitary representations $\rho: G \to U(H)$ have dimension at ...

**6**

votes

**2**answers

446 views

### Kazhdan's property (T) vs. residual finiteness

I have asked this question already on mathstackexchange but got no answer (see http://math.stackexchange.com/questions/1795795/kazhdans-property-t-vs-residual-finiteness) and it was suggested that I ...

**2**

votes

**0**answers

99 views

### Orthogonality relations for unitary representations of infinite (finitely generated) groups

Let $G$ be a group, and consider the matrix elements of finite dimensional irreducible unitary representations of $G$ over $\mathbb{C}$ as functions $f:G\to \mathbb{C}$. If $G$ is finite, any two ...

**3**

votes

**0**answers

72 views

### Cubic Directed Cayley graphs of 2-generated torsion-free groups

Is there a torsion-free group $G=\langle x,y \rangle$ such that the directed Cayley graph $\Gamma=Cay(G,\{x,y,x^{-1}y\})$ contains a finite cubic induced subgraph? The vertex set of $\Gamma$ is $G$ ...

**3**

votes

**1**answer

511 views

### On progress towards inverse Galois problem over rationals

I have heard that Progress towards the Inverse Galois problem over $\mathbb{Q}$
is very well documented for sporadic groups ($M_{23}$ is the only case open) and for $PSL_n(q)$.
From where I can read ...

**25**

votes

**3**answers

1k views

### What's the supersymmetric analogue of the Monster group?

Bosonic string theory lives in 26 dimensions, and it gives a conformal field theory where the field is a map from a Riemann surface to $\mathbb{R}^{24}$. The Leech lattice $L$ is an even unimodular ...

**15**

votes

**1**answer

274 views

### Does the injection $\text{Aut}(F_n) \hookrightarrow \text{Aut}(F_{n+1})$ split?

Let $F_n$ be the free group on $n$ letters.
The question is as in the title: letting $i:\text{Aut}(F_n) \hookrightarrow \text{Aut}(F_{n+1})$ be the natural injection, does there exist a homomorphism $...

**2**

votes

**0**answers

88 views

### Computing the order of elements in a non abelian exterior square of a finite group

If we have an explicit group $G$, and we pick two elements $g,h \in G$, could we find the order of the element $g \wedge h \in G \wedge G$?
The best thing I could find is Theorem 1.1 in Ellis' Book (...

**2**

votes

**0**answers

37 views

### The Socle of locally nilpotent $p$-group infinte rank

The Socle (is the subgroup generated by the minimal normal subgroups of $G$) of abelian $p$-group of infinite rank has infinite rank. (The term “rank” in the sense of the Mal’cev special or ...

**4**

votes

**1**answer

92 views

### Primary invariants

This question is related to the earlier question which is in the given link:
Primary invariants of a finite group
Let $G$ be a finite group and $V$ a complex representation of degree $n$, and let $...

**3**

votes

**1**answer

123 views

### A more precise description of conjugation of semi-simple subgroups

Let $G$ be a semi-simple algebraic group over $\mathbb{Q}$, I would like to find an integer $d>0$ only depending on $G$ with the following property. For any two semi-simple $\mathbb{Q}$-subgroups $...

**7**

votes

**0**answers

115 views

### An infinite torsion group $G$ with finite type $K(G,1)$?

There is a famous open problem in group theory that asks:
Does there exist an infinite finitely presented torsion group?
The general belief being that such groups exist. I would like to know ...

**4**

votes

**1**answer

118 views

### About the Eigenvalues of Orthogonal Matrix plus Perturbation

Let $O$ be an orthogonal matrix, $O^T O = I$, thus its eigenvalues lie on the unit circle, $\lambda(O)=e^{i\theta}$. Furthermore, assume the form
$O = X Y$, where both matrices satisfy $X^2 = I$ and $...

**4**

votes

**1**answer

245 views

### About the conjugation of semi-simple subgroups

Let $G$ be a semi-simple algebraic group over $\mathbb{Q}$, I would like to find an integer $d>0$ only depending on $G$ with the following property. For any two semi-simple $\mathbb{Q}$-subgroups $...

**0**

votes

**0**answers

39 views

### Explicit description of fields with ramification conditions

Let us fix an algebraically closed field $k$ of characteristic 0. If I understood correctly, the Riemann Existence theorem guarantees us existence of the field (Galois-)extension, say $F$, of $k(t)$ ...

**2**

votes

**0**answers

98 views

### Chief factors and local formation

Every thing below is concerned with finite groups.
My question is about this paper
A class of groups is a collection $\mathcal{X}$ of groups with the property that
if $G \in \mathcal{X}$ and if $H \...

**3**

votes

**1**answer

120 views

### Is there a matrix representation of the permutation group whose character is the Markov trace?

Let $g$ be an element in the permutation group (symmetric group) $S_N$. Define the Markov trace of $g$ (denoted $\text{tr}_k g$) as
$$\text{tr}_kg = k^\text{number of cycles in $g$} ,$$
which depends ...