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

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155 views

### Hausdorff Dimensions of Limit set of subgroups of SL(2,Z)

In a recent paper by Bourgain, Sarnak, Gamburd [1] talks about subgroups of $SL(2,\mathbb{Z})$.
Let $\Lambda$ be a finitely generated non-elementary subgroup of $SL(2,\mathbb{Z})$ with Hausdorff ...

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votes

**4**answers

368 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 ...

**5**

votes

**0**answers

89 views

### presentations of subalgebras

Assume that I have a finitely presented algebra $A$ over the complex numbers (by which I mean that $A$ is generated over $\mathbb{C}$ by finitely many elements $x_1,...,x_n$ subject to finitely many ...

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votes

**1**answer

106 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

130 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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**0**answers

128 views

### Automorphism groups for free groups with action

Let $A$ be a (finitely generated) free group and $G$ be a (finitely generated) free $A$-group - that is, a group with an action of $A$, which is free in the category of groups with an $A$-action. ...

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**2**answers

122 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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**1**answer

196 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 ...

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votes

**1**answer

442 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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**0**answers

78 views

### Space of polynomially growing harmonic functions on a Lie group

There is a recent theorem by Kleiner (based on previous work by Colding & Minicozzi), that reads:
If $G$ is a finitely generated group of polynomial growth, then for every $d$ the space of ...

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votes

**1**answer

196 views

### A question on permutations

Given integers $1$ through $n$, let $S$ be set of ordering of integers that respect even alternating or reverse alternating permutations (https://en.wikipedia.org/wiki/Alternating_permutation) up to ...

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**0**answers

134 views

### divisible 2nd cohomolgy group $H^2(G,\mathbb{Z}G)$

Recall that a (nontrivial) abelian group $A$ is called divisible if the multiplication by $n$ is surjection $A\to A$ for all $n\in\mathbb{Z}_{>1}$.
My question is the following.
Is there a ...

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votes

**2**answers

305 views

### Automorphisms of finite order in $Out(\widehat{F_2})$

Let $\widehat{F_2}$ be the pro-$\ell$ completion of the free group of rank 2, where $\ell$ is some prime.
Every outer automorphism of $F_2$ induces an outer automorphism of $\widehat{F_2}$, hence an ...

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votes

**1**answer

352 views

### What is the normal closure of $GL_2(\mathbb{Z})$ inside $GL_2(\mathbb{Z}_\ell)$?

This weird problem popped up in my research:
Let $\ell$ be a prime. Is there a description of the smallest normal subgroup of $GL_2(\mathbb{Z}_\ell)$ containing $GL_2(\mathbb{Z})$?
Is there a ...

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**0**answers

75 views

### Poincare inequality for connected Lie groups

Let $G$ be a compactly generated second countable locally compact group, and let $\mu$ be a probability measure which is:
symmetric,
adapted (in the sense that there is no proper subgroup $H$ such ...

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**1**answer

97 views

### Inducing representations from the stabilizer of a partition

For each positive integer $i$, let $A_i$ be a fixed representation of the symmetric group $S_i$. I won't tell you exactly what $A_i$ is, but let's say that I have a very explicit description of its ...

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**0**answers

56 views

### Is the family of probabilities generated by a random walk on a finitely generated amenable group asymptotically invariant?

Is the family of probabilities $\mu^n$ (convolution) generated by a random walk $\mu$ on a finitely generated amenable group $G$ asymptotically invariant ($\|g\mu^n-\mu\|_{L^1}\to 0$ for any $g\in ...

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votes

**2**answers

189 views

### Products of elliptic isometries

A well-known property on groups acting on trees is:
Theorem: Let $T$ be a tree and $g,h \in \mathrm{Isom}(T)$ two elliptic isometries. If $\mathrm{Fix}(g) \cap \mathrm{Fix}(h) = \emptyset$ then ...

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votes

**1**answer

156 views

### The first Betti number of a finite covering space of a closed 3-manifold

Let $\tilde{M}\to M$ be a finite covering of closed 3-manifolds.
Is it possible that $\beta_1(\tilde{M})-1> [\pi_1(M):\pi_1(\tilde{M})](\beta_1(M)-1)>0$?
Here, ...

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**0**answers

90 views

### Linear vs smooth actions of finite groups on spheres, euclidean spaces and closed disks

I would like to know examples (with references, if possible) of the following:
(1) a finite group $G$ acting effectively and smoothly on a sphere $S^n$ (any $n$) but admitting no effective linear ...

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**2**answers

86 views

### Are there references for the properties of words formed in finite groups using L-systems? (In particular, the algae L-system.)

Let $G$ be a (finite) group, and $a, b \in G$ be any two elements. Consider the sequence defined by
\begin{eqnarray*}
s_0 &=& a, \\
s_1 &=& b, \text{and} \\
s_{n+2} &=& s_{n+1} ...

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**0**answers

169 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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**0**answers

214 views

### Proving that a subgroup is normal

This question is partly inspired by the recent question on measurability and the axiom of choice.
Suppose I come up with a way to define a subgroup of a group, in a way that involves "no arbitrary ...

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votes

**0**answers

92 views

### Cycles covering the edges of the graph corresponding to the Van Kampen diagram of a presentation of a group

Let the group $G$ have the presentation $\langle x_1, \dots, x_n \;|\; r_1, \dots, r_m \rangle$. Let $\Gamma$ be a labelled directed graph corresponding to Van Kampen diagram over the above ...

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**0**answers

40 views

### Just-not-nilpotent-by-compact quotient of a locally compact group

It is known (and not complicated to prove) that for a finitely generated not virtually nilpotent group $G$, we can pass to a quotient $G/N$ of $G$, such that the quotient is just not virtually ...

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**0**answers

109 views

### Cayley graphs with special subgraphs and some related problems

I asked some questions about finite Cayley graphs with special type of subgraphs which has been answered by Dear Prof. Godsil. It can be seen in the MO page with address:
Cayley graphs and its ...

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147 views

### Uniform sub-linearity of sub-additive functions on groups

Suppose $G$ is a finitely generated group and suppose $f: G \to \mathbb{R}$ is subadditive function, that is: $f(g_1\circ g_2) \leq f(g_1) + f(g_2)$. One example of such $f$ is the word length in some ...

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**0**answers

32 views

### General form of a matrix $M$ commutes with the unitary representation $U^{\otimes m},~ \forall U\in U(n)$ [migrated]

My question is about the general form of a $n^m\times n^m$ positive definite matrix $M$ where
$$[M,U^{\otimes m}]=0,~ \forall U\in U(n)$$
or in other words, M commutes with all members of the the ...

**3**

votes

**1**answer

204 views

### Are there enough additive permutations?

I am hoping to learn enough about additive permutations to help with a number theory problem. These permutations also have connections to difference sets, orthomorphisms, transversals, and other ...

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**0**answers

59 views

### Categorizing finitely presented groups by complexity

Let $G$ be a finitely presented group tagged with a specific presentation $G \cong F\{X\} / \{h_1, \dots h_k\}$.
Preliminary: consider two presentations to have the same presentation class if they ...

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votes

**1**answer

275 views

### Which subgroup order of the symmetric group is the most frequent?

Question: What is the most frequent order of subgroups of $S_n$?
More precisely: Let $a_k$ be the number of subgroups of $S_n$ with order $k$. What is the maximum of $a_k$?
This question came up ...

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votes

**2**answers

344 views

### Unsolvability of a Quintic and its link with “Simplicity” of $A_{5}$

This is a re-post from MSE (because I did not get the kind of answer I wanted even after offering a bounty).
At the outset I must mention that I don't have a fairly working knowledge of Galois Theory ...

**48**

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,
...

**11**

votes

**2**answers

502 views

### Your favorite papers on geometric group theory

I would like to improve my culture on geometric group theory, so I am interested in short and accessible papers on subjects related to this field but which are not always available in the classical ...

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**0**answers

47 views

### Antichains in subgroups up to group automorphism

A family of subgroups of a group G will denote a nonempty collection of subgroups closed under conjugation and further passing to subgroups.
In a Noetherian group (any ascending chain of subgroups ...

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votes

**1**answer

107 views

### Spliting of short exact exact sequences of partially ordered groups

Consider a short exact sequence of partially ordered groups
$$0 \longrightarrow H \stackrel{\alpha}{\longrightarrow} G \stackrel{\beta} {\longrightarrow} G/H \longrightarrow 0 ,$$ where $H$ is a ...

**0**

votes

**1**answer

82 views

### SO(3) transformation that produces a reflection [closed]

This came up doing some research in quantum information. Let us consider two orthogonal three-dimensional unit vectors $v$ and $w$
$v^T\cdot w=0$,
and the Householder transformation
...

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votes

**0**answers

428 views

### Is a non-trivial finite perfect group of order 4n? [migrated]

A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$, or equivalently, if any $1$-dimensional complex representation is trivial.
Question: Is a non-trivial finite perfect group of ...

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**0**answers

191 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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**4**answers

763 views

### Determining if a matrix is orthogonal

Let g be an element of $GL_n(\mathbb C)$. We know that there are orthogonal groups $O(\beta)=\{X\in GL_n(\mathbb C) \mid X^t\beta X=\beta\}$ for any $\beta$, invertible symmetric matrix. Though these ...

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**0**answers

87 views

### Bound on $g(n+1)/g(n)$ for Landau's function

I have read that for the Landau function $g(n)$ (http://mathworld.wolfram.com/LandausFunction.html), one knows that
$\lim_{n \rightarrow \infty} g(n+1)/g(n) = 1$
(stated, but not proved in "On ...

**4**

votes

**1**answer

109 views

### Restricted Burnside Problem: Lower bound nilpotency class

Let $p$ be a prime and let $F$ be a free group of rank $d\geq 1$.
Kostrikin [1] proved that the $d$-generated Burnside group $B=B(d,p)=F/F^p$
of exponent $p$ has a maximal finite quotient
...

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votes

**3**answers

295 views

### What is the motivation and purpose of the Floretion group?

When searching through the Oeis, I came across something called a floretion. Based on the context, it seems to be some sort of algebraic structure. I googled it and found nothing that explained their ...

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**2**answers

189 views

### PSL(2,p) as quotient of triangle groups

As a by-product of some Magma computations, I've observed that, for each prime $p$
such that ${\rm PSL}(2,\mathbb{F}_p)$ can be a quotient of the $(3,3,4)$-triangle group
(i.e. $p \equiv \pm 1 ...

**5**

votes

**1**answer

180 views

### Hyperbolic knot complement groups and relative dimension

Let $G$ be a the fundamental group of a hyperbolic knot complement.
Then $G$ is hyperbolic relative to a subgroup $P\cong \mathbb Z \oplus \mathbb Z$.
The knot complement has a $2$-dimensional spine ...

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vote

**2**answers

225 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 ...

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**0**answers

90 views

### Application of finding shortest paths on Cayley graphs

For a fixed integer number $m$, Consider Cayley graph defined by all m-cycles in Symmetric group $Sym(n)$.
I know that for $m=2$,
there are some applications of finding shortest paths (or distance ...

**2**

votes

**1**answer

120 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

**1**answer

139 views

### Chinese remainder theorem for cyclic subfactor planar algebras

This post was inspired by an exchange with the indian woman mathematician Ajit Iqbal Singh.
The chinese remainder theorem can be stated as follows:
Let $n_1, \dots, n_r \ge 2$ be positive integers ...

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**0**answers

72 views

### What is the diameter of the Cayley graph of $S_n$ w.r.t the generator $\{(12),(23), . . . ,(n − 1 n) \}$? [duplicate]

Given a symmetric group $S_n $ and the generator set : $S = \{(12),(23), . . . ,(n − 1 n) \}$ is there any closed form expression for the diameter of the Cayley graph generated by this set of ...