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

**3**

votes

**1**answer

184 views

### Normal intermediate subgroup and normal core

Let $G$ be a finite group and $H$ a subgroup.
The normal core of $H$ in $G$ is $core_G(H) := \bigcap_{g \in G}g^{-1}Hg$
Definition: $K$ is a normal intermediate subgroup of the inclusion $(H ...

**4**

votes

**0**answers

76 views

### Groups of operators between local unitaries and full unitaries

Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...

**7**

votes

**2**answers

178 views

### Modifying Dehn's algorithm to allow equal length replacements?

I'm an analyst trying to understand a certain class of finitely presented groups (one example is below) so it's quite likely this question is naive but I hope it is at least intelligible. Given a ...

**5**

votes

**1**answer

193 views

### Kaplansky's idempotent conjecture for Thompson's group F

Let $K$ be a field and $G$ be a torsion-free group. Kaplansky's idempotent conjecture states that the group ring $K[G]$ does not contain any non-trivial idempotent, i.e. if $x^2=x$ then $x=0$ or ...

**0**

votes

**1**answer

72 views

### Is a weakly separable group always Lindelöf?

By "weakly separable" I mean the notion for uniform spaces used by David Wigner and Lawrence Brown: a uniform space is weakly separable if any uniform cover has a countable subcover. For a topological ...

**3**

votes

**0**answers

217 views

### On the Groups of Order $(p^2+1)/2$

A few days ago I asked a question (Groups of order $p(p^2+1)/2$) about a finite group of order $p(p^2+1)/2$ and I got a lot of useful information about it. Thanks for the nice and very helpful ...

**5**

votes

**1**answer

185 views

### Boundaries of relatively hyperbolic groups

When the interior of an n-manifold $M$ has a pinched negative curvature metric of finite volume, then its fundamental group $\Gamma=\pi_1M$ is relatively hyperbolic relative to the parabolic groups ...

**5**

votes

**5**answers

711 views

### Groups of order $p(p^2+1)/2$

It seems that when $p>3$ is a prime, then each group of order $p(p^2+1)/2$ is abelian as I checked by Gap for small $p$. Is it true for each $p$?
Thanks for your answers

**15**

votes

**1**answer

450 views

### Is a left topological group which is a manifold a topological group?

Let $G$ be a left topological group, i.e. a topological space with group operation such that left multiplication $L_g : x \mapsto gx$ is continuous (but right multiplication and inversion are not ...

**2**

votes

**2**answers

121 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

145 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**

votes

**0**answers

649 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**

votes

**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**

votes

**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

226 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

114 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

144 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

166 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

162 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**

votes

**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**

votes

**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

156 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 ...

**1**

vote

**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

103 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**

votes

**0**answers

172 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

317 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

199 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

96 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 ...

**4**

votes

**3**answers

354 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

114 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

177 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

410 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

874 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 ...