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

**2**

votes

**1**answer

140 views

### Does index 2 subgroup imply bipartite Cayley graph?

Let $G$ be a finitely generated group and let $\Gamma=Cay(G, S)$ be the Cayley graph of $G$ with respect to some generating set $S$.
If there exists $S$ such that $\Gamma$ is bipartite, then $G$ has ...

**0**

votes

**0**answers

47 views

### Largest order of automorphism group on a rooted tree? [migrated]

MacArthur, Sanchez-Garcia, and Anderson have used the ratio of the order of $|Aut(G)|$ and $n!$ (i.e., order of $S_n$) as a normalized measure of the symmetries present in a graph.
I am working on ...

**9**

votes

**1**answer

307 views

### Finitely generated group with $\aleph_0<X_G<2^{\aleph_0}$ normal subgroups?

Let $X_G$ be the number of normal subgroups of a group $G$. Are there examples of finitely generated groups $G$ where it is consistent to have $\aleph_0<X_G<2^{\aleph_0}$ normal subgroups? Also ...

**2**

votes

**0**answers

211 views

### Are the homogeneous single chain subfactors, Dedekind?

Background: See here and there.
Recall that a subfactor is Dedekind if all its intermediate subfactors are normal.
A subfactor $(N \subset M)$ is Homogeneous Single Chain (HSC) if its lattice ...

**11**

votes

**1**answer

456 views

### Find finite groups $G\cong Aut(G)$

I become interested in this problem because $G\cong Aut(G)$ suggests a special symmetry in $G$.
From $G/Z(G)\cong Inn(G)$ we know complete group is the anewer for the simplest case, though this class ...

**4**

votes

**0**answers

68 views

### Automorphisms with an orbit “transversal” to a subgroup

Let $G$ be a finite group, $H$ be a subgroup of index $2$, $\phi\colon G\to G$ an automorphism. For $d\in \mathbb N$ let us say that $\phi$ is $d$-transversal to $H$ iff there is $h\in H$ such that
...

**0**

votes

**1**answer

103 views

### Hamiltonian 2-groups

This is a group theory question. I am preparing a research paper. One result brought my attention. I am wondering if you know some paper or book listed this result.
Let $G$ be a 2-group. Suppose ...

**1**

vote

**1**answer

179 views

### On the character degrees of a finite group with special structure

Let $G$ be a finite group such that $G$ has a normal subgroup $N$ of order $p(p^2+1)/2$, where $p>13$ is an odd prime and $p\ne 239$. Also $G/N\cong \text{PSL}(2,p)$. Can we say that there exists a ...

**4**

votes

**1**answer

204 views

### Subgroups of Gromov's hyperbolic groups

It's known that subgroups of Gromov's hyperbolic groups are not necessarily hyperbolic.
Is there any counter-example when the quotient is Abelian. More precisely, let $G$ be a Gromov's hyperbolic ...

**0**

votes

**0**answers

80 views

### Brauer characters of finite simple group $E_8(5)$

I would like to find the irreducible characters of the group $E_8(5)$ (mod 2)?
Can anyone help? (I am elementary in working with Brauer characters)
Many thanks

**1**

vote

**1**answer

235 views

### Odd subgroup of $\mathrm{GL}(n,\mathbb{Z})$

The group $\mathrm{GL}(n,\mathbb{Z})$ acts on $(\mathbb{Z}/2\mathbb{Z})^n$ by right multiplication (the same kind of things can be done with left action). I denote by $H\subset ...

**2**

votes

**0**answers

206 views

### Fundamental group of a Cayley graph [migrated]

Could someone please indicate the proof of the following fact (I believe strongly, but I am not sure, that this is true). Let $F$ be a free group of finite rank, $N<F$ a normal subgroup and ...

**2**

votes

**1**answer

68 views

### Stabilizer of a nonsingular vector in a quadratic space (char (k)=2)

suppose that $k$ is a finite field of characteristic 2 and $(V,q)$ a quadratic space, i.e., $V$ is a $k$-vector space and $q:V\to k$ quadratic form. Suppose that $\dim(V)\geq 4$ and that $q$ is ...

**12**

votes

**2**answers

585 views

### Infinitely many finitely generated groups having the same Cayley graph

Is there an unlabeled locally-finite graph which is a Cayley graph of an infinitely many non-isomorphic groups with respect to suitably chosen generating sets?

**3**

votes

**1**answer

219 views

### A little bit of Intuition for Corepresentations from Representations

I asked this question over on Math.Stack --- where it has a bounty --- but I didn't really get a helpful response so I am asking the question here. One commenter suggests that I am confusing left- ...

**0**

votes

**0**answers

86 views

### Dehn's Fundamental Problems

For what sort of groups are the Dehn's Fundamental Problems solvable,that is for what sort of groups are the word,conjugacy and isomorphism problems solvable.

**23**

votes

**2**answers

921 views

### Fractal-like structures arising from the action of a group on $\mathbb{Z}^2$

Let $G := \langle a, b, c \rangle < {\rm Sym}(\mathbb{Z}^2)$ be the group
generated by the permutation
$$
a: \ (m,n) \ \mapsto \ (m-n,m)
$$
of order $6$ and the involutions
$$
b: \ (m,n) \ ...

**1**

vote

**1**answer

151 views

### List of Conjugacy Classes of the General Linear group over $\mathbb{Z}/p^2\mathbb{Z}$

Does anybody know the answer to this or a good way to go about working this out? I have a list for $GL_2(Z/pZ)$ and I am trying to lift it to this; I have mostly been using fairly elementary algebraic ...

**1**

vote

**0**answers

104 views

### Conditions for a semidirect product to be a complete group

Let A, B and H be groups with $\theta_{1}$ an injective homomorphism from H to Aut(A) while $\theta_{2}$ is an homomorphism from H to Aut(B).
We know there is a subgroup of Aut(A $\times$ B) which ...

**-5**

votes

**1**answer

146 views

### on the solvable groups of order $p^aq^b$ [closed]

We know that if $ p$ is a prime number then $ O^p (G) $ is the smallest normal subgroup of $ G $ such that $ G/O^p (G) $ is a $ p $-group.
Now let $ G $ be a finite group of order $ p^aq^b $ where $ ...

**2**

votes

**2**answers

133 views

### Group of homomorphisms with real coefficients and circle coefficients [closed]

Do you know some groups $G$ that have the same group of homomorphisms to $R$ and to $S^1$; i.e. $$Hom(G,R) = Hom(G,S^1)??$$ Is there any special property for $G$ in order to satisfy the last relation?
...

**2**

votes

**0**answers

190 views

### The tallest possible lattice?

Let O be a complete discrete valuation ring and G a finite group. Recall that a finitely generated O-free OG-module $M$ such that the traces of the invertible endomorphisms of $M$ generate a strictly ...

**3**

votes

**0**answers

86 views

### Groups such that all finite-dim representations are finitely presented

Let $G$ be an infinite group. What sorts of finiteness properties can I put on $G$ to ensure the following holds for all $M$?
Let $M$ be a finite-dimensional vector space over $\mathbb{Q}$ upon ...

**15**

votes

**2**answers

374 views

### Is there a natural notion of completion of a Coxeter system?

Let $(W,S)$ be a Coxeter system. Then any element of $W$ can be written as a finite products of elements of $S$. I want some notion of a "completion" of $W$, call it $\hat{W}$, whose elements are ...

**9**

votes

**0**answers

288 views

### Does anybody know if the Fourier algebra of SL(3,Z) has an approximate identity?

(Note to those who like to tidy LaTeX, or ${\rm \LaTeX}$: I kindly request that you don't put any LaTeX in the title of this question, nor change the bolds below to blackboard ...

**4**

votes

**1**answer

141 views

### Which hyperbolic tilings are Cayley graphs?

I realise the question is easy but after asking to a few people (and never getting a clear answer), I thought it could be instructive to ask it here:
Given a regular tiling of the hyperbolic plane is ...

**1**

vote

**0**answers

138 views

### Centralizers of elements in HNN extensions

Let $G$ be a (countable) group, $H<G$ be a proper subgroup and $\theta:H\to G$ be an injective group homomorphism such that $\theta(H)\neq G$. Consider the HNN extension $\Gamma=<G,t \mid ...

**5**

votes

**1**answer

335 views

### Finite groups for which the element orders form an arithmetic progression

Which are the finite groups $G$ such that the element orders of $G$ form an arithmetic progression? Several remarks:
$S_3$, $A_4$ and any $p$-group of exponent $p$ satisfy this property.
If $G$ ...

**6**

votes

**1**answer

275 views

### Using math software to show that the following groups are infinite?

I would like to show that the following finitely presented group in 3 generators $P, Q, R$ is infinite in certain cases:
$$P^p, Q^q, R^r, (PQ)^2, (QR)^2, (PQR)^2, (QR^{r/2+1})^a (RQR^{r/2})^b$$
For ...

**2**

votes

**1**answer

118 views

### Proving that a closed walk of some odd length k exists on a Cayley graph

I'm trying to prove the following:
Let $G$ be a group with finite symmetric generating set $S$ and let $\Gamma(G,S)$ be the corresponding Cayley graph. Let $X_1, X_2,\cdots$ be a simple random walk ...

**4**

votes

**0**answers

255 views

### More on Moonshine for the Thompson group and weakly holomorphic weight one half modular forms

This question is a follow-up to
Monstrous Moonshine for Thompson group $Th$?
and is based on various comments to that question, in particular S. Carnahan's mention of the
connection to known ...

**2**

votes

**1**answer

318 views

### A second isomorphism theorem for the inclusions of groups

The usual second isomorphism theorem for groups is: let $G$ be a group, $S$ and $N$ subgroups with $N$ normal, then $SN$ is a subgroup of $G$, $S\cap N$ is a normal subgroup of $S$ and $SN/N \simeq ...

**3**

votes

**1**answer

80 views

### Siegel domains and cuspidal functions

Let $F$ be a number field and $\mathbb{A}$ the ring of adeles over $F$. We consider $P_{n}$ the mirabolic subroup of $GL_{n}$.
Do we have a analog of Siegel subset for the quotient ...

**7**

votes

**2**answers

443 views

### What is the name of this type of groups?

Suppose $A$ is a finite set and $\Sigma=A\cup A^{-1}$. Let $L\subseteq \Sigma^{\ast}$ be a regular language on the alphabet $\Sigma$. Is there a common name for the group $G$ presented as:
$$G=\langle ...

**13**

votes

**2**answers

584 views

### Groups which are only defined up to conjugation

I'm trying to understand what the right way is to think about "groups which are only well-defined up to conjugation." Since this is somewhat vague let me clarify it by pointing out the main examples ...

**4**

votes

**1**answer

138 views

### Can Haar measure fail to be bi-invariant without conjugation shrinking a set?

(This is a slightly reformatted and clarified version of my question from math.SE, since I believe
the answer there is wrong and its poster has not responded to my comment in over two weeks.)
Let ...

**3**

votes

**0**answers

168 views

### Groups whose finite index subgroups are isomorphic

I am looking for examples of the following situation:
$G$ is an infinite group. Every two finite index proper subgroups of $G$ are isomorphic.
The only examples that I have now are (1) ...

**2**

votes

**1**answer

206 views

### on the prime divisors of $(p^2+1)/2 $

The following question is equivalent to a problem in group theory.
Let $ p > 13$ be a prime number distinct from 239. Let $ a=(p^2+1)/2 $. Is there any prime divisor $r$ of $a$ such that $r\mid ...

**4**

votes

**1**answer

245 views

### Locally finite compact groups

I assume all tolpological groups here to be Hausdorff. A group is called locally finite if every finitely generated subgroup is finite. What can be said about a locally finite compact group? Must it ...

**3**

votes

**1**answer

179 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

75 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

177 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

191 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

216 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 ...

**4**

votes

**1**answer

182 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

710 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

447 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

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
${\mathrm{N}}_{{\mathrm{GL}}_2({\Bbb R})}({\mathrm{SL}}_2({\Bbb Z}))$?