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

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3
votes
1answer
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
0answers
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
2answers
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
1answer
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
1answer
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
0answers
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
1answer
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
5answers
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
1answer
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
2answers
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
4answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
2answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
2answers
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
0answers
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
1answer
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
1answer
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
2answers
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
1answer
202 views

Candidates for non-sofic groups

What are the "simplest" examples of countable groups that are not known to be sofic?
9
votes
3answers
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
2answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
2answers
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
2answers
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
1answer
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
2answers
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
1answer
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
1answer
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
3answers
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
0answers
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
1answer
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
1answer
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
4answers
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
3answers
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 ...