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

**4**

votes

**0**answers

71 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 ...

**6**

votes

**1**answer

544 views

### Positivity of the alternating sum of indices for boolean interval of finite groups

Let $G$ be a finite group and $H$ a subgroup such that the interval $[H,G]$ is a boolean lattice.
Let $L_1, \dots , L_n$ be the maximal subgroups of $G$ containing $H$.
Let the alternative sum ...

**9**

votes

**2**answers

625 views

### Folner sets and balls

Several related questions were asked before on MO, but it is not clear to me if the following was settled.
Given a finitely generated amenable group, is it always possible to find some finite ...

**3**

votes

**0**answers

50 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 $r∈R$. ...

**12**

votes

**2**answers

447 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 ...

**11**

votes

**0**answers

121 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 ...

**1**

vote

**0**answers

50 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 ...

**8**

votes

**1**answer

832 views

### A dual version of a theorem of Øystein Ore in group theory

Let $(H \subset G)$ be an inclusion of finite groups.
This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved:
Theorem: $\mathcal{L}(H\subset G)$ ...

**0**

votes

**0**answers

67 views

### Finding stable ideals of $\mathbb{F}_3[[X,S]]$ by group action

Let $k > 1$ be a positive integer and define the action $\sigma_k$ on $\mathbb{F}_3[[X,S]]$ by:
$\sigma_k: X \mapsto X + S + X^k$
$\sigma_k: S \mapsto S + S^3$.
Conjecture: There exists a ...

**4**

votes

**0**answers

125 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 ...

**-3**

votes

**0**answers

79 views

### Finding Bijection between Permutation Set [on hold]

$\beta$ is a subset of symmetric group $S_n$ which acts on $n$ elements of set $X$. Permutations of
$\beta$ acts on $k $ elements of $X$ only.
Set $L$ is a set of $n$ labels which labels ...

**3**

votes

**1**answer

218 views

### A question on $p$-central $p$-groups

Let $p$ be a fixed prime. A group $G$ is termed $p$-central if every element of order $p$ in $G$ lies in the center.
Having a finite $p$-group $G$ of rank $k$ (the least integer, such that every ...

**2**

votes

**2**answers

121 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 ...

**4**

votes

**2**answers

210 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 $C_2\times A_4$?

**2**

votes

**0**answers

81 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 ...

**11**

votes

**2**answers

1k views

### Generalization of a theorem of Øystein Ore in group theory

Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive.
Proof: see below.
Let $(H \subset G)$ be an inclusion of finite groups and ...

**5**

votes

**3**answers

434 views

### On the theory of infinite extraspecial $p$-groups

$p$ is a prime number. A group $G$ is called an infinite extraspecial $p$-group if
1) it is infinite,
2) every $g\neq 1$ in $G$ has order $p$,
3) its centre $Z(G)$ coincide with $G'$ and is a ...

**7**

votes

**1**answer

193 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 ...

**3**

votes

**0**answers

58 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 ...

**1**

vote

**0**answers

90 views

### Presentation of hyperbolic groups [on hold]

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

**-5**

votes

**0**answers

41 views

### Homomorphism, Group Theory [closed]

Let G=Z4, the group of integers modulo 4, and let H be the Klein four group, let f: G->H be a homomorphism. Why does the kernel of f must contain the element of 2 of G?

**1**

vote

**1**answer

191 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 $\mathrm{SO}(4)$?

**13**

votes

**2**answers

234 views

### Schur multiplier of $Sp(2g, \mathbb{Z}/2)$ for $g \geq 3$

This question is about the computation of $H_2(Sp(2g, \mathbb{Z}/2), \mathbb{Z})$, where $Sp(2g, \mathbb{Z}/2)$ is the group of symplectic $2g \times 2g$ matrices over $\mathbb{Z}/2$.
With respect to ...

**6**

votes

**2**answers

380 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

185 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 ...

**0**

votes

**2**answers

356 views

### Mean value theorems for the Haar integral?

Let $G$ be a compact topological group (feel free to add hypotheses if necessary). Is there any mean value theorem for its (normalized to 1) Haar integral?
In general, are there mean value theorems ...

**23**

votes

**2**answers

2k views

### Order of products of elements in symmetric groups

Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying
$1 < a, b, c \leq n-2$ the symmetric group ${\rm S}_n$ has elements of order $a$ and $b$
whose product has order ...

**2**

votes

**0**answers

89 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

**1**answer

456 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 ...

**3**

votes

**0**answers

56 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 $G$ ...

**-4**

votes

**0**answers

44 views

### what is the Relationships between metric space and set Topology [closed]

state the Relationships between metric space and set Topology
Explain with examples
If the relationships has any significance

**24**

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 ...

**3**

votes

**1**answer

114 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 ...

**13**

votes

**2**answers

803 views

### Dehn's algorithm for word problem for surface groups

For some $g \geq 2$, let $\Gamma_g$ be the fundamental group of a closed genus $g$ surface and let $S_g=\{a_1,b_1,\ldots,a_g,b_g\}$ be the usual generating set for $\Gamma_g$ satisfying the surface ...

**4**

votes

**1**answer

239 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 ...

**3**

votes

**2**answers

336 views

### The number of subgroups of ${\frak S}_n$

Because of my interest in this question, I listed the subgroups of ${\frak S}_n$ for $1\le n\le4$. I found that the number of subgroups are, respectively, $1,2,6,24$. It might be a coincidence, or it ...

**14**

votes

**1**answer

251 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

76 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

86 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

162 views

### Must normalizing field outer automorphisms “divide” the dimension?

Imprecise question: To get a normalizing field outer automorphism of
order $r$, must we multiply the dimension by $r$?
Precise hypothesis: Let $p\geqslant 5$ be a prime, let $q$ be a power of $p$ and ...

**4**

votes

**1**answer

108 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 ...

**12**

votes

**2**answers

298 views

### When are growth series rational?

For a finitely generated group $G$ with generating set $S$, one can define the word metric. Using this, we are able to define the sequence $\sigma(n)$ by
$ \sigma(n) = |\mathcal{S}(e,n)| $, which is ...

**6**

votes

**3**answers

556 views

### Index of derived subgroup in derived group

Let $G$ be a group with finite index subgroup $H$. Let $G^\prime = [G,G]$ denote the derived subgroup of $G$.
Is it true that $|G:H|<\infty$ implies that $|G^\prime: H^\prime|<\infty$.
If this ...

**6**

votes

**3**answers

423 views

### Subgroups of finitary symmetric groups

Question 1: Does there exist an intrinsic characterization of groups $G$ isomorphic to some subgroup of some finitary symmetric group (i.e. all the permutations of a given set that fix
all but ...

**7**

votes

**0**answers

106 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

**2**answers

215 views

### A finiteness property for semi-simple algebraic groups

Let $G$ be a semi-simple algebraic group over a field $K$, I am considering a question about whether there exists a finite set of semi-simple $K$-subgroups, say $H_1,...,H_r$, such that for any ...

**0**

votes

**0**answers

38 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 $k(t)$ ...

**3**

votes

**1**answer

111 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 ...

**2**

votes

**0**answers

95 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 ...