# Tagged Questions

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

**2**

votes

**2**answers

120 views

### Normal subgroups of automorphism group of infinite cubic tree

Let $T$ denote the unique (countably infinite) tree in which every vertex has degree $3$ and let $G=Aut(T)$ be the group of graph automorphisms of $T$.
I'm interested in the properties of this ...

**4**

votes

**4**answers

553 views

### Normalizer of SL_2(Z) in GL_2(R)

What is the normalizer of ${\mathrm{SL}}_2({\Bbb Z})$ in ${\mathrm{GL}}_2({\Bbb R})$? Namely
${\mathrm{N}}_{{\mathrm{GL}}_2({\Bbb R})}({\mathrm{SL}}_2({\Bbb Z}))$?

**1**

vote

**1**answer

142 views

### subgroups of General linear group with two generators

In General linear group $GL(n,q)$, every matrix is conjugate with its rational canonical form. Using this fact we have the conjugacy classes of cyclic groups. My question is about the groups which are ...

**21**

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**0**answers

640 views

### Monstrous Moonshine for Thompson group $Th$?

I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2,
$$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, ...

**6**

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**0**answers

96 views

### How much do the classes of geodesic metrics and Green metrics on a Gromov hyperbolic group differ?

Background
Let $G$ be a Gromov hyperbolic group. If $G$ acts properly discontinuously and cocompactly on a proper geodesic metric space $X$, then any bijective identification of $G$ with its orbit ...

**0**

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**0**answers

114 views

### Is the direct product of distributive inclusions of groups, modular?

Let $H$ a subgroup of $G$ and $\mathcal{L}(H \subset G)$ the lattice of intermediate subgroups ($\mathcal{L}( G)$ if $H= \{ e \}$).
Definitions: A lattice $(L, \wedge, \vee)$ is :
- Distributive if ...

**0**

votes

**0**answers

69 views

### Groups and triangle-square complexes

I would like to know what kind of groups (and/or their group presentation) acting geometrically on CAT(0) curved piecewise Euclidean triangle-square complexes.
Thanks

**0**

votes

**2**answers

118 views

### Products of maximal inclusions of finite groups with a non-obvious intermediate

Let $(H_1 \subset G_1)$ and $(H_2 \subset G_2)$ be core-free maximal inclusions of finite groups.
Their product, the inclusion $(H_1 \times H_2 \subset G_1 \times G_2)$, admits four obvious ...

**0**

votes

**1**answer

224 views

### Centralizer of derived subgroup

In all questions suppose $G$ metabelian p-group such that
G is not regular ( so $cl(G) \geq p$ ), G is not a wreath product;
$Z(G) \leq \phi(G)$.
1) Let $M$ normal abelian subgroup of $G$ such ...

**1**

vote

**1**answer

113 views

### About the power subgroups of the Bianchi groups

Let $B$ be the Bianchi group and let $B^2 =\langle x^2 | x \in B\rangle$, the power subgroup of $B$.
Is it true that $B \ne B^2$ all the time ?

**1**

vote

**1**answer

143 views

### JSJ-decompositions of hyperbolic groups and elementary vertices

My question is the following:
In Bowditch's JSJ-decomposition of hyperbolic groups, can elementary (virtually-cyclic) vertices have degree 1? If not, why not?
I had thought for a long time that ...

**2**

votes

**0**answers

165 views

### Existence of inclusions of finite groups with a particular lattice property

Definition : Let $\sim$ be the equivalence relation on inclusions of finite groups, generated by :
$(H \subset G) \sim (\phi(H) \subset \phi(G))$, with $ \phi: G \to L$ a finite group morphism and ...

**3**

votes

**1**answer

158 views

### Permutation groups transitive on partitions into ordered pairs

The following came up in a problem on graph reconstruction. It isn't very important, but I thought some people here might find it interesting and not too trivial (I'm not a group theorist).
Take a ...

**3**

votes

**2**answers

161 views

### Universal Central Extension of pi(X), X a compact Riemann surface of genus>1

Does a universal central extension exist for the fundamental group of a Compact Riemann Surface of genus1? Please give a detailed explanation.I am unable to justify the statements in Atiyah-Bott Phil ...

**2**

votes

**0**answers

68 views

### Subgroups with minimal centralizers

Inspired by this question Subgroups with trivial Centralizers, one can define a characteristic subgroup in any group $G$ as follows
$$\Lambda(G)=\bigcap\{ H\leq G: C_G(H)=Z(G)\}$$
Is there any ...

**0**

votes

**1**answer

182 views

### Subgroups with trivial Centralizers

Let $G$ be a compact, simply-connected, and simple Lie group. Let $H$ be a closed subgroup of $G$ that has the same centralizer as the center of $G$. Is there a nice classification of such subgroups?
...

**2**

votes

**1**answer

125 views

### Non-exchangeable unimodular graph

Let $G=(V,E)$ be an countably infinite, locally finite transitive graph. Say that $G$ is exchangeable if for every two vertices $v,w \in V$ there exists a graph homomorphism that maps $v$ to $w$ and ...

**6**

votes

**2**answers

195 views

### Uniform-in-p classification* of p-groups of order p^n for each fixed n?

To what extent is there/can there be a description that is uniform in p (for p sufficiently large) of the p-groups of order $p^n$, for each fixed n?
Note 1: I used the word "description" rather ...

**5**

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**1**answer

202 views

### Candidates for non-sofic groups

What are the "simplest" examples of countable groups that are not known to be sofic?

**9**

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**3**answers

418 views

### Classification of automorphism groups of groups of order $p^4$

For the purpose of classifying another algebraic structure which is parametrised by the choice of a group and of an automorphism I need the classification up to isomorphism of automorphism groups of ...

**7**

votes

**2**answers

183 views

### 3-cocycle representatives for the dihedral group $D_{2n}$?

I am looking for a reference for a complete list of 3-cocycle representatives for $H^3(D_{2n},\mathbb{C}^\times)$, where
$$
D_{2n}=\langle a, b\mid a^2=b^2=(ab)^n=e\rangle
$$
is the dihedral group of ...

**3**

votes

**0**answers

154 views

### vertex transitive and Cayley graphs

(all the graphs alluded to below are finite).
Suppose I gave you a graph, and asked you whether it was vertex-transitive. How hard is that algorithmically?
The second question is: suppose I gave you ...

**1**

vote

**1**answer

111 views

### A formula for isotropy group $\pi_1(G_a)$

Let $G$ be a compact Lie group and $T$ be its maximal tours, and $a\in \mathfrak{g}^*$. and $G_a$ be the isotropy group of $G$ then $T\subset G_a$ and we know that $\pi_1(T)=\mathbb{Z}^n$. My ...

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**0**answers

116 views

### How decomposable a modular representation can you get by reducing a given p-adic representation?

General Background
Take $G$ to be a finite group, and say $V$ is s $G$-representation over $\mathbb{Q}_p$. By picking a $G$-invariant lattice $L\subset V$ we can get an $\mathbb{F}_p$ representation ...

**2**

votes

**1**answer

196 views

### Maximal centralizers in General linear group

I will be so thankful for any comments or help about the following question.
Is it possible to obtain all maximal centralizers in the General linear group $GL_n(F)$, for an arbitrary field $F$ and ...

**3**

votes

**0**answers

101 views

### Computing the pro-solvable closure of a finitely generated subgroup of a free group

The pro-solvable topology on a group $G$ is the unique group topology such that the set of normal subgroups $N\lhd G$ with $G/N$ a finite solvable group is a fundamental system of neighborhoods of the ...

**2**

votes

**1**answer

174 views

### Do all right orderable groups have the Haagerup property?

Do all right orderable groups have the Haagerup property?
Recall that a group is right orderable if there exists a total order $\leq$ on it such that $a\leq b\Rightarrow ac\leq bc$. This property is ...

**7**

votes

**0**answers

105 views

### Fusion pattern in a cyclic subgroup of order 8

Can a finite simple group $G$ have an element $x$ of order 8 such that $x^2$ is conjugate to $x^{-2}$ but $x$ is not conjugate to any element of $\{x^3,x^5,x^7\}$?
In other words, does this fusion ...

**3**

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**0**answers

170 views

### Invariants of groups that are invariant under passage to finite index subgroups

This question is mostly idle curiosity.
Recall the following standard terminology: if $P$ is a property of groups, a group $G$ is said to be virtually $P$ if it has a subgroup of finite index which ...

**3**

votes

**2**answers

316 views

### A structure of the group of automorphisms of an infinite binary tree

My friend asked me to ask his question here. Where he can find (a paper or a book) containing a complete description (with the proof) of a structure of the group of automorphisms of an infinite binary ...

**3**

votes

**2**answers

211 views

### Suslin's Stability Theorem for Chevalley Groups

I am looking for a version of Suslin's Stability Theorem for Chevalley groups.
The version of the theorem for $G=SL_n({\mathbb Z}[x_1, \dots , x_m])$ states that the if $n\ge m+2$, the elementary ...

**2**

votes

**1**answer

198 views

### Does every nearly normal subgroup contain a normal subgroup?

Let $H$ be an infinite subgroup of a discrete group $G$. $H$ is called nearly normal if it is commensurable with a normal subgroup $K$ of $G$, that is $H\cap K$ is a finite index subgroup of both $H$ ...

**5**

votes

**2**answers

321 views

### When is a subgroup of a Lie group itself a Lie group?

Assume that $G$ is a Lie group. Is it understood which subgroups of $G$ are Lie groups?
Ideally, I would like to make no extra assumption about $G$. In particular, $G$ can be infinite dimensional.
...

**1**

vote

**1**answer

95 views

### Index of agemo subgroups in $p$-groups

Having a finite $p$-group $G$ ($p$ odd). we denote by $\Omega_1(G)$ the subgroup generated by all the elements of $G$ of order dividing $p$.
Is there an example of such a group $G$, such that ...

**1**

vote

**1**answer

345 views

### Abelian subfactors, a relevant concept?

Through the questions below, this post asks whether the concept of abelian subfactor is relevant.
Remark : here abelian qualifies an inclusion of II$_1$ factors $(N \subset M)$, $N$ is not an abelian ...

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votes

**3**answers

353 views

### Jordan-Hölder theorem for subfactors?

All the subfactors $(N\subset M)$ are irreducible and finite index inclusions of II$_1$ factors.
First recall that in this paper, D. Bisch characterizes the Jones projections $e_K$ of the ...

**3**

votes

**0**answers

113 views

### Examples of divisible Lie algebras

We say that a nonzero Lie algebra $L$ is divisible, if for all elements $a$ and $b$ with $a\neq 0$, there exists $x\in L$ such that $[a, x]=b$. What are examples of divisible Lie algebras?

**1**

vote

**1**answer

184 views

### $SO(N^2-1)$ and the adjoint representation of $SU(N)$

It is a known fact that the adjoint representation of $SU(N)$ is a proper subgroup of $SO(N^2-1)$.
I would like to know how a generic $(N^2-1)\times (N^2-1)$ special ($det =1$), orthogonal matrix $O$ ...

**8**

votes

**1**answer

175 views

### Is there a finitely generated residually finite group with solvable word problem that does not embed in a finitely presented residually finite group?

The famous Higman embedding theorem says that every recursively presented group embeds in a finitely presented group. This is a convenient tool to construct finitely presented groups with bizarre ...

**10**

votes

**4**answers

409 views

### Iterated semi-direct products

Let $G$ be a finite group. Suppose that we can write $G= A \rtimes B$ and also $A = C \rtimes D$. Further suppose that C is normal in $G$ (not just in $A$). Then can we write $G = C \rtimes E$ where ...

**21**

votes

**3**answers

873 views

### How can classifying irreducible representations be a “wild” problem?

Let $q$ be a prime power and $U_n(\mathbb{F}_q)$ be the group of unitriangular $n\times n$-matrices. I've read and heard in several places (see e.g. this mathoverflow question) that classifying ...

**5**

votes

**1**answer

101 views

### Uniqueness of composition series for profinite groups

(It is possible that an answer to this question can be found in the literature, but I couldn't find anything after searching for about an hour.)
Let $G$ be a compact, totally disconnected, second ...

**5**

votes

**1**answer

173 views

### Jordan-Hölder theorem for planar algebras?

First recall the Jordan-Hölder theorem for groups:
Theorem (Jordan-Hölder): Let $G$ be a group, and let $$ G=G_1 \supset G_2 \supset \dots \supset G_r = \{ e \} $$ be a normal tower such that ...

**13**

votes

**1**answer

278 views

### An infinite, amenable, finitely presentable group with no non-trivial finite quotients

My question is a simple one: is there a group with the properties in the title?
In the absence of the 'finitely presentable' hypothesis, an example is provided by Juschenko--Monod's construction of a ...

**0**

votes

**0**answers

65 views

### About the product of a subset and its inverse set

Let $G$ be a group and $A$ its nonempty subset. Denote by $A^{-1}$ the set of inverses of its elements. It is obvious tthat $A$ is a subgroup iff $A^{-1}A=A$ iff $A^{-1}A\subseteq A$ iff $AA^{-1}=A$ ...

**5**

votes

**3**answers

356 views

### Finite index free subgroups of $\mathrm{SL}(3,\mathbb{Z})$

Does $\mathrm{SL}(n,\mathbb{Z})$ have a free subgroup of finite index for some $n \geq 3$? I know that $\mathrm{SL}(3,\mathbb{Z})$ has many free subgroups and that in the case of ...

**2**

votes

**2**answers

240 views

### Is any continuous group homomorphism from R to C* an exponential map? [closed]

Consider $\mathbb{R}$ to be an additive topological group, and $\mathbb{C}^{\ast}$ to be a multiplicative topological group.
Is the following statement true?
If so, then how can one prove it?
...

**2**

votes

**1**answer

206 views

### The category of subfactors extending the category of groups?

This post was inspired by this answer of Dave Penneys.
In the category of (irreducible hyperfinite II$_1$) subfactors, the morphisms of $(N \subset M)$ to $(N' \subset M')$ are usually defined as ...

**3**

votes

**1**answer

195 views

### An encryption scheme using properties of non-abelian groups

I had a discussion about various encryption schemes with a colleague yesterday, and the following thought came to my mind: is it possible to devise an encryption scheme exploiting the phenomenon that ...

**3**

votes

**2**answers

202 views

### Factor subset of finite group

Let $G$ be a group of order $n$ and $d$ a positive divisor of $n$. Is it true that there exists a subset $A$ of $G$ with $d$ elements and a subset $B$ such that $G=AB$ and $|AB|=|A||B|$ ...