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

**-4**

votes

**0**answers

55 views

### Isomorphism between $\mathbb R^3$ and the the Heisenberg group

What is the isomorphism between $\mathbb R^3$ and $\mathbb C \times \mathbb R$ as a group (the Heisenberg group), provided with the law
$$(z,t).(w,s) = (z+w, t+s+\Im m(z\bar{w})\, ); \quad z,w\in ...

**0**

votes

**0**answers

64 views

### Generating-bijective groups

We may say that two finitely generated groups $G$ and $H$ are generating-bijective when there exist homomorphisms $\phi:G\rightarrow H$ and $\psi:H\rightarrow G$ such that, for each ordered generating ...

**0**

votes

**0**answers

31 views

### Free cocompact action of discrete group gives a covering map [migrated]

I'd like to find a short proof of the following seemingly basic fact encountered on the second page of Atiyah's paper "Elliptic operators, discrete groups, and von Neumann algebras." ...

**6**

votes

**0**answers

131 views

### When does a CAT(0) group contain a rank one isometry

Let $G$ be a CAT(0) group which acts on the CAT(0) space $X$ properly and cocompactly via isometry. Let $g \in G$ be a hyperbolic isometry of $X$. Then $g$ is called $\textbf{rank one}$ if no axis of ...

**2**

votes

**0**answers

126 views

### For $P$ $\mathbb{Z}G$-projective, $\mathbb{Q}\otimes P$ is $\mathbb{Q}G$-free

I'm looking for a proof of a theorem of Swan [1, Theorem 3]:
If $G$ is a finite group and $P$ a finitely generated projective $\mathbb{Z}G$-module, then $\mathbb{Q}\otimes_\mathbb{Z}P$ is a free ...

**-1**

votes

**0**answers

67 views

### An efficient algorithm for computing all semigroups of order n [on hold]

I attached two papers which give an algorithm for computing all semigroups of order n=3, and n=5.
I understood the first(table 3) and second(table 4) steps of algorithm, but I can't understand the ...

**1**

vote

**1**answer

113 views

### Number of eigenvalues of a Cayley graph

Let $G=Z_2^n$ and $S\subset G$. Is there any relation for number of distinct eigenvalues of $\Gamma=Cayley(G,S)$ graph depending on $n$ and $|S|$, or at least diameter of $\Gamma$? If you have any ...

**0**

votes

**0**answers

97 views

### Rank of a locally free $\mathbb Z[G]$-module

This is a basic question. Let $G$ be a finite group, $M$ a finitely generated $\mathbb Z[G]$-module so that the $\mathbb Z_p[G]$-module $M_p$ is free for all prime numbers $p$, i.e. is locally free.
...

**19**

votes

**1**answer

491 views

### Does $E_8$ know $Spin(7)$?

One way to define the compact group $Spin(7)$ is as the stabilizer of a certain 4-form on Euclidean $\mathbb R^8$ (see e.g. this MO question). This 4-form can be defined in various ways. For ...

**4**

votes

**1**answer

161 views

### A finiteness property for semi-simple algebraic groups

Let $G$ be a semi-simple algebraic group over a field $K$, I am considering a question about whether there exists a finite set of semi-simple $K$-subgroups, say $H_1,...,H_r$, such that for any ...

**2**

votes

**0**answers

447 views

### Is this group given presentation isomorphic to $\mathbb{Z}_2$, and why? [migrated]

We have the group $G = \langle a, \,b \; | \; a^2b^2ab^{-1}, \, a^3b^4a^{-2}b^{-3}\rangle$.
Obviously $G/G' = \mathbb{Z}_2$. Is it true that $G = \mathbb{Z}_2$ or equivalently that $G'=1$? It is ...

**4**

votes

**1**answer

209 views

### Weil's Haar measure construction from below

Weil's construction of a Haar measure on a locally compact group rests on approximating a function from above by sums of translates of another function.
I would need to know something similar for an ...

**6**

votes

**0**answers

98 views

### Constructing the largest finite group with a fixed number of conjugacy classes

It is known that there are finitely many finite groups with a given number of conjugacy classes. How can one construct (or get a character table for) the groups $G$ that realize the maximum possible ...

**0**

votes

**0**answers

64 views

### Is $n+\frac {1}{2}$ in Kendall-Mann numbers and quantum harmonic oscillator related?

It is known that quantum harmonic oscillator is connected to the symmetric group of infinite order which is isomorphic to the permutation group. According to Cayley's theorem any finite group is ...

**5**

votes

**0**answers

134 views

### Hemi-Semi Direct Product

In the category of racks (similarly quandles), instead of well-known semi direct product, we have hemi-semi direct product construction as seen on Wagemann & Crans.
As far as I know, semi direct ...

**3**

votes

**1**answer

86 views

### Extension property for unipotent linear groups over rings

This is my first question, so my apologies if it is too simple/poorly motivated.
During the course of some recent research I came across a particular variant of the following problem.
Let $G$ ...

**3**

votes

**0**answers

50 views

### Characteristically simple locally compact abelian groups

Say a topological group $G$ is topologically characteristically simple if there does not exist a closed subgroup $1 < K < G$ such that $K$ is invariant under all automorphisms of $G$ (here ...

**3**

votes

**1**answer

131 views

### Are braid groups conjugacy separable?

I would like to re-ask a question that was raised in the comments here:
Normal subgroups of braid groups
Namely, a group $G$ is called conjugacy separable, if it holds that for every two elements ...

**4**

votes

**0**answers

77 views

### Finitely presented amenable LERF group which is not virtually solvable

Is there a group $G$ with the following properties?
Finitely presented
Amenable
Not virtually solvable
LERF (that is, every finitely generated subgroup is closed in the profinite topology on $G$).
...

**5**

votes

**2**answers

472 views

### A question about (unicity of certain cycles in a Cayley graph of a) symmetric group

Let $S=\{(1,2),(1,2,3,\ldots,n),(1,2,3,\ldots,n)^{-1}=(1,n\ldots,2)\}$ be a subset of the symmetric group $S_n$. We know that $(1,2,\ldots,n)(1,2)=(2,3,\ldots,n)$, and thus ...

**9**

votes

**2**answers

270 views

### Lower central series quotients in terms of (co)homology

Let $G$ be a group. It is well-known that $H_1(G,\mathbb{Z})=G/[G,G]$. Also (at least up to torsion) $[G,G]/[G,[G,G]]=\Lambda^2H^1(G,\mathbb{Z})/H_2(G,\mathbb{Z})$ as explained, for example, in this ...

**3**

votes

**1**answer

155 views

### Properties of a special finitely presented groups

Recently, when I was working with Cayley graphs, I faced up with a special group. The original group is as follows:
$$G:=<a,b,c|ab=ba,a^{10}=cbc^{-1}>.$$
We can show that this group can be ...

**3**

votes

**0**answers

67 views

### Limits of quotient groups in the space of marked groups

In the space of marked groups with $m$ generators, suppose that a sequence $(G_i, S_i)$ converges to $(G, S)$. For any $i$, let $K_i$ be a normal subgroup of $G_i$ and assume that $\bar{S}_i$ is the ...

**10**

votes

**1**answer

247 views

### About positive upper density

For $S\subset \mathbb{N}$ define the upper density as $D^{\ast
}(S)=\limsup_{n\rightarrow \infty }\frac{\left\vert S\cap \{1,2,\ldots,n\} \right\vert }{%
\left\vert n\right\vert }.$
Question: ...

**3**

votes

**1**answer

164 views

### Is there a characterization of CI-groups of order less than 100?

We know some benefit criterion in articles such as:
C. H. Li, On isomorphisms of finite Cayley graphs-a survey, Discrete Math., 256 (2002) 301-334.
C. H. Li, Z. P. Lu, P. ...

**2**

votes

**1**answer

174 views

### Is there a formula for the number of elements in $S_n$ having length $k$ with respect to the generators taken to be the transpositions?

Define the length of a permutation as the minimum number of adjacent transpositions needed to describe it.
We know that there is only one element of length $0$ in $S_n$ and $n-1$ elements of length ...

**13**

votes

**0**answers

306 views

### Does the category of G-spectra know G?

I was recently in the situation of having access to the category of $G$-modules (for some group $G$ which I had forgotten), as just a category, i.e. no monoidal structure, together with the forgetful ...

**3**

votes

**0**answers

87 views

### A finite supersolvable group with generators of prescribed order

Let $G=\langle a,b\rangle$ be a finite supersolvable group. Is there any special information about the structure of $G$ when $o(a)=2$ and $ o(b)=2^k > 2$?

**3**

votes

**1**answer

96 views

### What is known about semigroups that are generated by (cyclic) subgroups?

A semigroup $(A,\cdot)$ that is idempotent (i.e. $a^2=a$ for every element $a\in A$) is naturally generated by its subgroups (every element on itself constitutes a trivial group). I would like to know ...

**5**

votes

**6**answers

452 views

### Transitive permutation groups which all of their proper subgroups are intransitive

Let $G$ be a transitive permutation group on a finite set $\Omega$. It is clear
that if $G$ is regular, then every proper subgroup of $G$ is intransitive. Is
there any other class of groups with this ...

**6**

votes

**1**answer

392 views

### What is this Lie algebra?

Consider two matrices $A,B \in \mathfrak{su}(N)$ which are both diagonal in the standard basis and non-zero.
If we consider the new matrix $\tilde{B} := FBF^{\dagger}$ where $F$ is the `quantum' ...

**6**

votes

**1**answer

138 views

### Morita equivalence base equivalence relation for discrete groups

In the context of "discrete groups", is there an equivalence relation that implies the Morita equivalence of their reduced group C*-algebras?
We define $G \sim H$ for discrete groups $G$ and $H$, ...

**17**

votes

**3**answers

385 views

### Conjugacy classes of $SL_2(Z)$

I was wondering if there is some description known for the conjugacy classes of $\{A\in GL_2(\mathbb{Z})|\;\;|Det(A)|=1\}$ or $SL_2(\mathbb{Z})$. I was not able to find anything about this. Most ...

**3**

votes

**1**answer

125 views

### Torsion-free, normal subgroups of certain Coxeter groups

Let $G$ be the reflection group of a regular, 4-dimensional, hyperbolic honeycomb. I would like to find a family $H_i < G$ of finite-index, torsion-free subgroups of $G$, so that I can represent ...

**-1**

votes

**0**answers

94 views

### Is $(\mathbb R^d/\mathbb Z^d,+)$, $d>2$, isomorphic to some group of an algebraic surface? [migrated]

It is a well-known fact that for points of a cubic curve over $\mathbb{RP}^2$ we can define a group $(G_{\mathbb{RP}^2},+)$ using Cayley–Bacharach theorem.
See Wiki: The group law.
Another fact ...

**1**

vote

**0**answers

68 views

### An optimal lower bound related to generators in a boolean interval of finite groups

Let $[H,G]$ be a rank $n$ boolean interval of finite groups (i.e. $[H,G] \simeq B_n$ as lattice).
Let the set $E = \{ g \in G \ | \ \langle H,g \rangle = G \}$
Remark: If $g \in E$ then $Hg ...

**5**

votes

**0**answers

77 views

### Random pro-p groups via iterated uniformly random central extensions

Inspired by this question on math.se, I want to understand the following construction of a random pro-$p$ group:
We want to construct an inverse system
$$\cdots \xrightarrow{\alpha_i} G_i ...

**1**

vote

**1**answer

128 views

### Some question on haar measure for sumsets of closed subsets of profinite groups

Let $H$ be a profinite group with the haar measure $\mu_H$. Let $H_1$ and $H_2$ be closed subgroups of $H$. $H_1$ and $H_2$ have their own haar measures $\mu_{H_1}$ and $\mu_{H_2}$ respectively.
...

**11**

votes

**1**answer

179 views

### commutators in upper triangular matrices

Consider the group $T_p(n)$ of all non-singular upper triangular matrices with entries in $\mathbb{F}_p.$ Its commutator subgroup is $U_p(n)$ (all elements in $T_p(n)$ with $1$s on the main diagonal). ...

**4**

votes

**1**answer

181 views

### Does there exist finite dimensional irreducible representation of Euclidean or Poincare group in which translation and rotation both act nontrivially?

Does there exist any finite dimensional irreducible rep. of Euclidean or Poincare group in which translation and rotation both act nontrivially?
Let me firstly clarify my question. For example, we ...

**5**

votes

**0**answers

113 views

### Lifting automorphisms of quotient groups

I am concerning here a natural question:
Problem: Let $G$ be a finite group, and let $N$ be a characteristic subgroup of $G$. When can an automorphism $\varphi\in\mathrm{Aut}(G/N)$ be lifted to an ...

**3**

votes

**0**answers

140 views

### The homotopy type of the mapping space $Map_{B\rho}(BS^1,BG)$? for $G$ a compact Lie group

Given a homomorphism $\rho:S^1\rightarrow G$ with $G$ a compact Lie group there is an induced map of classifying spaces $B\rho:BS^1\rightarrow BG$. What is known about the homotopy type of the mapping ...

**8**

votes

**0**answers

177 views

### How bad can $SK_1$ of a commutative ring be?

For a commutative ring $R$ define $\mathrm{SK}_1(n, R)=\mathrm{SL}(n, R)/\mathrm{E}(n, R)$, the quotient of the special linear group by its subgroup generated by the elementary matrices. When ...

**7**

votes

**1**answer

228 views

### Generic set that is a proper subgroup

For a group $G$ generated by a finite set $S$ we denote by $B_{G,S}(n)$ the ball of radius $n$, that is the set of all elements in $G$ which are expressible as products $x_1x_2\ldots x_n$ where ...

**1**

vote

**1**answer

179 views

### Cohomologically trivial $G$-modules

Is there a finite non-abelian $2$-group $G$ without non-trivial elementary abelian direct factor and of order $2^9$ satisfying the following condition: $$Z(G) \cap Z(\Phi(G))= \langle ...

**4**

votes

**0**answers

70 views

### Abnormal subnormal series

Consider a group $G.$ Is it possible for $G$ to have a subnormal series $G \triangleright G_1 \triangleright \dotsc \triangleright G_n \triangleright \dotsc$ which cycles - that is, with $G_{i+k} ...

**3**

votes

**2**answers

113 views

### Finite solvable groups with abelian Fitting subgroup

Let $G$ be any finite solvable group with Fitting subgroup
$F(G)$. Which conditions on $F(G)$ makes $G$ to be supersolvable?
(It is well-known that any finite solvable group with cyclic Fitting ...

**15**

votes

**5**answers

464 views

### Cayley graphs of $A_n.$

Consider the Cayley graphs of $A_n,$ with respect to the generating set of all $3$-cycles. Their properties must be quite well-known, but sadly not to me. For example: what is its diameter? Is it an ...

**3**

votes

**1**answer

190 views

### Braid groups over the sphere

Can any one give me an example of surjective homomorphism on braid groups on the sphere that is not injective?
Such that $B_{n}(S^2)$ is generated by $\sigma_1,\sigma_2, \dots, \sigma_{n-1}$ which are ...

**2**

votes

**1**answer

149 views

### 2-closure of a permutation group

Let $G$ be a group acting on a set $\Omega$ faithfully. Then 2-closure of $G$ denoted by
$G^{(2)}$ is the largest subgroup of the symmetric group of $\Omega$ with
the same orbits as $G$ on ...