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

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-4
votes
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
2answers
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
2answers
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
6answers
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
1answer
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
1answer
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
3answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
2answers
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
5answers
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
1answer
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
1answer
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 ...