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

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14
votes
0answers
325 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
93 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
111 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
470 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
401 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
139 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 $H$, ...
16
votes
3answers
417 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
128 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
vote
0answers
71 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
79 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
131 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. ...
10
votes
1answer
188 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
194 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 ...
4
votes
0answers
131 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
145 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 ...
7
votes
0answers
240 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 $n\...
7
votes
1answer
244 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 $x_i\...
2
votes
1answer
186 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 \prod_{i=1}^{2^d}...
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
123 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
481 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
195 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
166 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 $\Omega\...
6
votes
1answer
325 views

Liouville property - a very basic question

Let $\mathbb{F}_2$ be the free group on two generators. By a result of Kaimanovich and Vershik, for each measure $\mu$ on $\mathbb{F}_2$ such that the support of $\mu$ generates $\mathbb{F}_2$, we ...
2
votes
0answers
166 views

Unique product groups (and semigroups)

A group $G$ is called a u.p.-group (short for unique product group) if for all nonempty finite subsets $A,B\subseteq G$, there exists an element $g\in A \cdot B$ which can be uniquely written as a ...
6
votes
2answers
186 views

Finding an “optimal” quotient in a free group

Consider the abelian free group $G = \mathbf{Z}^n$ of rank $n$ and a finite subset $A \subset G \setminus \{0\}$. Since $G$ is residually finite, there is a subgroup $H \subset G$ such that $A \cap H =...
7
votes
3answers
237 views

Signs in Chevalley's commutator formula

I am trying to understand presentations of twisted groups of Lie type (specifically $^2D_5$) over finite fields using Steinberg's presentations (for instance from Gorenstein, Lyons and Solomon, Number ...
4
votes
0answers
132 views

Groups with infinitely many finite conjugacy classes

I've been coming across the condition "IMFCC: having infinitely many finite conjugacy classes" often in recent times and I was wondering if there is any serious difference between having "IMFCC" and ...
2
votes
1answer
103 views

Intransitive finite irreducible linear groups whose orbits are all large

I am interested in intransitive irreducible linear subgroups $G\subseteq\mathrm{GL}_n(\mathbb{F}_p)$ acting on $V-\{0\}=\mathbb{F}_p^n-\{0\}$ in the natural way, such that all of the orbits are very ...
6
votes
2answers
257 views

Cardinality of factors of infinite non-abelian groups

Let $A$ and $B$ be arbitrary nonempty subsets of a group $G$. Then the product $AB$ is called direct, and we denot it by $A \cdot B$, if the representation of every its element by $x=ab$ with $a\in ...
10
votes
0answers
291 views

A question concerning model theory of groups

Several days ago, Professor Martin Bridson gave a very nice talk in my department. Some questions concerning his talk came into my brain Since I am neither a model theorist nor a algebraist, I am not ...
9
votes
0answers
159 views

Detecting a module for the free group algebra on a finite quotient

Let $F_2$ be the free group on two generators $x,y$ and let $R$ be the group algebra $\mathbf{Q}[F_2]$. Let $a,b,c$ be integers. Then define a right $R$-module $M = R / (ax + by + c) R$. I am ...
7
votes
2answers
266 views

Thompson group $V$

R. Thompson introduced three groups $F\subset T\subset V$. The question concerning amenability of $F$ is still unanswered and has attracted much attention. I have read that Thompson group $V$ contains ...
13
votes
1answer
336 views

Cohomology of central extensions of groups

Let G be a central extension of a finite group H by $Z/2$. I need an explicit description of the differentials $d_2$ and $d_3$ in the Lyndon-Hochschild-Serre spectral sequence which converges to the ...
4
votes
0answers
95 views

How to enumerate a discrete group of matrices by their Frobenius norm?

Suppose I have a discrete group $G<\mathrm{SL}_2(\mathbb{C})$, and it is finitely generated by some known generators. That is, $G=\langle g_1,\dots,g_n\rangle$. The Frobenius norm of a matrix $m=\...
1
vote
1answer
190 views

What are the finite subgroups of $\operatorname{Sp}_{2n}(\mathbb{Z})$?

I've read the following question: Finite subgroups of ${\rm SL}_2(\mathbb{Z})$ (reference request) and it made me wonder. It's easy to see that $\operatorname{SL}_2(\mathbb{Z})=\operatorname{Sp}_2(\...
3
votes
1answer
158 views

Braid group: Can a left-twist increase the number of right twists?

Disclaimer: This question was first posted on math.se without any answer. This is something that naturally occurs in my research, but I am no expert on this - it feels like a natural question so I am ...
0
votes
1answer
126 views

The relationship between $p$-solvable Group and solvable group [closed]

Can anyone please tell me The relationship between $p$-solvable Group and solvable group.and find an example of a $p$-solvable group that is not solvable group or vice-versa.
7
votes
0answers
222 views

A “direct” proof that hyperbolic groups are not amenable

I am looking for a proof that a finitely generated hyperbolic groups is non-amenable [unless it is virtually cyclic] which is as "metric/combinatoric" as possible. Here are the two proofs I am aware ...
4
votes
1answer
118 views

A decomposition of $w_0$ which is similar to the reduced decomposition

Some basic definitions about reduced decomposition: In the symmetric group $S_n$, let $s_i$ denote the adjacent transposition $(i,i+1),i\in \{1,2,\cdots,n-1\}.$ Since $S_n$ is generated by adjacent ...
5
votes
1answer
171 views

A question on the commutativity degree of the monoid of subsets of a finite group

The commutativity degree $d(G)$ of a finite group $G$ is defined as the ratio $$\frac{|\{(x,y)\in G^2 | xy=yx\}|}{|G|^2}.$$It is well known that $d(G)\leq5/8$ for any finite non-abelian group $G$. If $...
1
vote
2answers
106 views

Any analysis on phase of eigenvalue of unitary matrix?

I understand that there are invariant Haar measure for eigenvalues of unitary matrix. I further understand that absolute value of eigenvalues of unitary matrix is 1. But, I could not find any analysis ...
0
votes
2answers
516 views

Is the absolute Galois group the same as the automorphism group? [closed]

Is the absolute Galois group $\mathrm{Gal}(\overline{\mathbf{Q}}|\mathbf{Q})$ the same as the group $\mathrm{Aut}_{\mathbf{Q}}(\overline{\mathbf{Q}})$ the automorphism group in the category of $\...
2
votes
2answers
439 views

For a ring R, does $GL_n(R)$ embed into $GL_m(F)$ for some field F?

Suppose $R$ is a noncommutative ring. What is the sufficient and necessary condition for $GL_n(R)$ embedding into $GL_m(F)$ for some field $F$? In particular, what if $R$ is the group ring $\mathbb{Z}...
0
votes
0answers
78 views

Embedding a cancellative monoid into another in such a way that $|X-x|=|X|$, where $X$ is a fixed finite set and $x\in X$

Preliminaries. Let $\mathbb A = (A, +)$ be a possibly non-commutative semigroup. For $X, Y \subseteq A$ we set $$ X - Y := \{a \in A: a + y \in X\text{ for some }y \in Y\}, $$ which is just the usual ...
3
votes
1answer
99 views

free subgroups of the fundamental group of nonorientable surfaces

Sorry for this possibly trivial or stupid question, but I'm very far from being an expert in Algebra. Let G be the fundamental group of the nonorientable surface of even rank n=2k (n generators, 2n ...
6
votes
1answer
196 views

Embedding a cancellative monoid into another in such a way that a prescribed element becomes left-invertible

Let $\mathbb A = (A, +_A)$ be a cancellative, but possibly non-commutative, monoid with identity $0$, and fix an element $x \in A$. Does there always exist a cancellative monoid $\mathbb B = (B, +)$ ...
7
votes
1answer
244 views

Does the Fano plane “embed” in the complex projective plane?

$PSL(2,7)$ acts on the projective plane over $\mathbb{F}_2$ (the Fano plane) through its identification with $GL(3,2)$. It also acts on the projective plane over $\mathbb{C}$ through either of its ...
1
vote
0answers
124 views

Which conjectures are proved for sofic groups? [closed]

Which conjectures about groups are resolved in case of sofic groups? I know two examples: Kaplansky's direct finiteness conjecture (proved by Gabor Elek). Some versions of Ornstein's isomorphism ...
31
votes
1answer
562 views

Identities of commutators

Let $G$ be a group and set $[x,y]:= x^{-1}y^{-1}xy$ as usual and consider it as a binary operation. Question: Is there a description of the identities that the operation $[.,.]$ satisfies for all ...