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

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4
votes
0answers
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
1answer
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
2answers
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
0answers
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
2answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
2answers
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
2answers
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
0answers
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
2answers
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
3answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
2answers
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
2answers
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
0answers
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
2answers
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
2answers
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
0answers
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
1answer
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
0answers
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
0answers
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
3answers
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
1answer
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
2answers
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
1answer
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
2answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
2answers
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
3answers
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
3answers
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
0answers
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
2answers
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
0answers
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
1answer
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
0answers
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 ...