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Questions tagged [gr.group-theory]

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

1
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1answer
14 views

Transitive embedding of the projective space $P^2\Bbb R$ into the $4$-sphere

Is there an embedding (i.e. injective continuous map) $$\phi:P^2\Bbb R\hookrightarrow S^4\subseteq\Bbb R^5$$ of the 2-dimensional projective space $P^2\Bbb R$ into the $4$-sphere, that is ...
0
votes
0answers
27 views

Connected transitive group action and wreath product

It is well known and not hard to prove that the wreath product of two finite transitive group actions is again transitive. Apparently a stronger statement is true: suppose that $G$ and $H$ act ...
0
votes
0answers
28 views

Coset representatives of a principal congruence subgroup by another principal congruence subgroup

Consider the principal congruence subgroup $\Gamma(N)$, this consists of entries in $\mathrm{SL}_2(\mathbb{Z})$ congruent to the identity modulo $N$. Consider another principal congruence subgroup $\...
2
votes
0answers
129 views

What are the points about representation of groups? [on hold]

For a fixed (let say finite-, or Lie-, to respect the historical motivations) group, why does the study of all its linear representations over a fixed field, leads to some knowledge about its ...
12
votes
1answer
225 views

Traveling Salesman Problem on finite group

Given a finite group $H$, define a norm on $H$ to be a function $f : H \rightarrow \mathbb{R}_{\geq 0}$ satisfying: $f(x) = 0 \iff x = e$ is the identity; $\forall x \in H$, we have $f(x) = f(x^{-1})$...
1
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0answers
109 views
+100

Centralizers of Cartan subgroup III

Let $\mathcal O$ be an order in an imaginary quadratic field $K$. Let $n$ be a positive integer. The multiplicative group $(\mathcal O/n\mathcal O)^\times$ acts on the module $\mathcal O/n\mathcal O\...
8
votes
1answer
333 views

Cohomology of simple finite groups remembers the group?

Let $G$ and $H$ be finite simple groups. I expect that if $G$ and $H$ are not isomorphic, then their cohomology groups with integral coefficients are not all isomorphic, that is, $H^*(G,\mathbb{Z})...
6
votes
2answers
207 views

Subalgebra of a group algebra

Let $k$ be a field, $G$ a finite group, and $k[G]$ the group algebra. Let $A$ be a subalgebra of $k[G]$. In general, $A$ is not the group algebra of some subgroup $H$ of $G$. Question: Is there any ...
3
votes
0answers
94 views

Liftings and splittings (reference request)

I'm writing a paper and, at a certain point, I need the following, rather elementary Lemma. Assume that we have a commutative diagram of short exact sequences of groups of the form Then the ...
2
votes
2answers
233 views

Is the size of a conjugacy class in a finite classical group a polynomial?

Suppose $G$ is a classical matrix group over a finite field of order $q$. If $C$ is a conjugacy class in $G$ , is $|C|$ a polynomial in $q$? This question is supported by the fact that whenever I ...
1
vote
1answer
285 views

Variation on Sylow Theorems [closed]

Does a finite group on $2^t$ elements (with $t$ a positive integer) necessarily have a subgroup of index two? It seems close to the Sylow Theorems but not quite. Maybe there is a simple ...
3
votes
1answer
133 views

Group cohomology of $S_3$ in terms of its Sylow subgroups

I am trying to understand $H^*(S_3, M)$ in terms of it's Sylow $p$ subgroups. From III.10.2 and III.10.3 in Brown we know that \begin{equation}H^n(G,M) = \bigoplus_p H^n(H,M)^G\end{equation} where $p$...
3
votes
0answers
104 views

Has the external knit product been used to construct a previously unknown group?

In the Wikipedia article Zappa–Szép product , the knit product (a.k.a. Zappa–Szép product, Zappa–Rédei-Szép product, general product, exact factorization) is defined, and its basic properties are laid ...
-1
votes
0answers
91 views

Formal sums on groups with symmetric finite presentations [closed]

Suppose we have a group $G$ with a "symmetric" finite presentation: \begin{equation} G = \langle g_i, i=1\ldots m | R_\alpha (\{g_i\}) ,\, \alpha=1\ldots n \rangle \end{equation} \begin{equation} R_\...
-1
votes
0answers
81 views

n-abelian group

Suppose that $G$ is a free group generated by a set $X$, $G=F\{X\}$. We define a homomorphism of groups $a_{n}:G\rightarrow G$ such $a_{n}(x)=x^{n}$ for any element $x\in X$. there is an obvious ...
0
votes
0answers
41 views

Absolute center of finite $p$-group

For any group $G$, the absolute center $L(G)$ of $G$ is defined as $$L(G) = \lbrace g\in G\mid \alpha(g)=g,\text{ for all }\alpha\in Aut(G) \rbrace,$$ where $Aut(G)$ denote the group of all ...
1
vote
0answers
52 views

About $E(G)$ for a finite $p$-group $G$

For any group $G$, the absolute center $L(G)$ of $G$ is defined as $$L(G) = \lbrace g\in G\mid \alpha(g)=g,\forall\alpha\in Aut(G) \rbrace$$, where $Aut(G)$ denote the group of all automorphisms of $G$...
21
votes
0answers
319 views

Do two integral matrices generate a free group?

Is it decidable whether two given elements of ${\rm GL}(n,{\mathbb Z})$ generate a free group of rank 2? This is a simple question that I have been asking people for the past couple of years, but ...
3
votes
1answer
106 views

Regular semisimple elements in $SL(n,q)$

Consider $G=GL(n,q)$. A regular semisimple element of this group, is a matrix, whose characterestic polynomial is square-free and same as minimal polynomial. Results show that the number of such ...
0
votes
0answers
71 views

Finding Elusive Orbits in GL action on polynomials

I am attempting to generate orbit representatives for the action of $\operatorname{GL}(n, F_2)$ on homogeneous polynomials of fixed degree $d$ in $n$ variables using random methods. However, some ...
2
votes
0answers
76 views

A theory of (or reference for) symmetric point arrangements

I wonder where I can find something written on symmetric point arrangements (see definition below). I am interested in general references, preferably books that introduce (or papers that use) some ...
3
votes
1answer
117 views

Lawvere metrics on the poset of subgroups of Z?

Background: Recall that a Lawvere metric structure on a set $X$ consists of a function $d\colon X\times X\to[0,\infty]$ satisfying two properties: $d(x,x)=0$ for all $x\in X$, $d(x,y)+d(y,z)\geq d(x,...
1
vote
0answers
94 views

Relation between groups $A_n$, $B_n$, $D_n$ and $S_n$ or inversions of random elements in Coxeter groups

First of of all I'm trying to find a general interpretation to the following facts below. Let's look at the property of Kendall-Mann numbers $M(n)$ which are row maxima of Triangle of Mahonian ...
5
votes
0answers
91 views

Group with Character Degrees {1,pq,pr,qr}, where p,q and r are distinct primes

I am currently trying to bound the derived length of certain solvable groups assuming that they have only two irreducible monomial complex character degrees. Using induction, it often suffices to ...
24
votes
1answer
1k views

Have the Quantum Group Theorists taught the Group Theorists Anything?

I will start with the general before moving to the specific. Consider for a moment the two (very) soft definitions. An abstraction of an object $X$ is a category $\mathcal{C}_0$ such that $X$ ...
2
votes
1answer
84 views

Lie algebra of an algebraic group generated by connected subgroups

Let V be a vector space over an algebraically closed field. Let $\{H_i\}_{i \in I}$ be a collection of closed connected subgroups of $\operatorname{GL}(V)$ (wrt. Zariski topology). It is a basic ...
2
votes
0answers
66 views

Unitary matrices $p$-root of identity such that the Fourier transform matrices are $p$-root of identity

Take a prime number $p$ and $\omega=e^{2i\pi/p}$. Assume we have p complex matrices (in finite dimension $n$) $A_0, \dotsc, A_{p-1}$ such that $\forall i, A_i^p=I$. Define the $p$ fourrier transform ...
6
votes
0answers
145 views

Automorphism group of poset of number fields

Consider the poset of number fields, partial order being defined by inclusion of fields. What is the group of order-preserving automorphisms of this poset? What if we take only Galois extensions of $\...
0
votes
0answers
42 views

Trying to understand pointwise convergence in the symmetry group of X [migrated]

Apologies for a question that I'm sure is very silly, but I'm not an expert on symmetry groups and this example is confusing me. So, suppose we define the following functions from $\mathbb{R}\...
8
votes
1answer
157 views

Unitary representations of finite groups over finite fields

I would like to learn the basic theory of unitary representations of finite groups over finite fields. Here, the unitary group $\operatorname{GU}(n,\mathbb{F}_{q^2})$ consists of all invertible ...
-1
votes
1answer
129 views

Groups With Arbitrarily Large Torsion [closed]

Thompson's Group has two well known presentations: $\langle x_0,x_1, ... $ | $ x_k^{-1} x_n x_k = x_{n+1}\forall k < n \rangle$ $\langle A,B $ | $ [AB^{-1}, A^{-1}BA], [AB^{-1}, A^{-2}BA^2] \...
7
votes
1answer
134 views

Number of words of length N that reduce to the identity in a specific Coxeter group

Suppose we have a Coxeter group whose diagram is given by a simplex. In other words, $G=\langle g_1,\ldots ,g_k\mid(g_i)^2=e,\,(g_ig_j)^3=e \rangle$. How many words of length $N$ simplify to the ...
5
votes
1answer
153 views

“Dimension” of discrete subgroups of infinite covolume in Lie groups

Let $G$ be a semisimple Lie group with finite center, $K$ a maximal compact subgroup, and $d=\dim(G/K)$. Let $\Gamma$ be a non-cocompact discrete subgroup of $G$. [Edit: assume that $\Gamma$ is ...
1
vote
0answers
277 views

Braided lobsters

If $(X,m)$ is a median algebra, then for each $x\in X$, define an operation $\wedge_{x}$ by letting $y\wedge_{x}z=m(x,y,z)$. Then $(X,\wedge_{x})$ is a meet-semilattice with least element $x$. Define ...
2
votes
1answer
83 views

Ends of G-spaces with action of a finitely generated group

This question is a development of my previous question. Let $G$ be a finitely generated group acting transitively on an infinite set $X$ so that for every $g\in G$ and $x\in X$ the $g$-orbit $\{g^nx:...
2
votes
0answers
59 views

Centralizers of Cartan subgroups II

Let $K$ be an imaginary quadratic field and let $\mathcal O$ be its ring of integers. Suppose that $2$ is split in $\mathcal O$. Let $k$ be a positive integer. The multiplicative group $(\mathcal O/2^...
3
votes
0answers
102 views

Existence of loxodromic elements in certain subsets of $\text{PSL}_2(\mathbb C)$

Let $R$ be a subset of $\text{PSL}_2(\mathbb C)$ and consider its natural action on $\mathbb {CP}^1$. We say that $R$ is elementary if either $R$ is conjugated to a subset of $\text{SU(2)}$ or if ...
2
votes
0answers
64 views

Faithful group actions on tuples from algebraic structures

So I am looking for examples of the following phenomenon. Suppose that $V$ is a variety with a computable equational theory which is not locally finite. Suppose that $G$ is an infinite finitely ...
6
votes
0answers
134 views

A group of all whose elements are distorted

Does there exist a finitely presented group, not torsion, all of whose infinite-order elements are distorted? An infinite-order element $g$ of a finitely generated group $G$ is undistorted if there ...
6
votes
2answers
233 views

Ends of finitely generated torsion groups

It is known that the number of ends of a finitely generated group is 0,1, 2 or $\infty$. Problem 1. What is known about the number of ends of infinite finitely generated torsion groups? In ...
2
votes
1answer
175 views

Malcev's paper “On a class of homogeneous spaces” in English

I am struggling to find the English translation of Malcev's paper "On a class of homogenous spaces" providing foundational material for nil-manifolds. To be precise this paper: Malcev, A. I. On a ...
2
votes
1answer
76 views

A Backtrack as a Single Word in a Group Presentation yields a Complex that isn't of the Same Homotopy Type?

By "backtrack" I mean a subword of a relator in a group presentation of the form $x x^{-1}$. Let $X = \langle a \rangle$ as a presentation complex. Let $Y = \langle a$ | $aa^{-1} \rangle$ as a ...
2
votes
1answer
123 views

Centralizers of Cartan subgroups

Let $E$ be an elliptic curve with CM by an order $\mathcal O$ in an imaginary quadratic field $K$. Choose a basis for $E[N]$ to get an isomorphism $\operatorname{Aut}(E[N])\cong \operatorname{GL}_2(\...
7
votes
1answer
132 views

Classification of the Extraspecial 2-groups $H_n$

I have a sequence of groups $H_n$ which I know to be extraspecial 2-groups of order $2^{2n+1}$. I also know the number of order 4 elements I have in $H_n$ for every $n$. Precisely, the number of order ...
6
votes
0answers
121 views

Short exact sequence of free topological groups

Suppose that $K\rightarrow G \rightarrow G/K $ is a short exact sequence of topological groups such that $G$ and $G/K$ are free topological groups. Is it true that we have a continuous section $s: G/K\...
4
votes
1answer
109 views

Unipotent completion of free group

Whilst I am reading articles on unipotent completion to understand its basic construction, I found something confusing. Let $F$ be a free group of rank 2 whose generating letters are $x$ and $y$ and ...
5
votes
1answer
150 views

Volume of balls in homogeneous manifolds

Let $X=G/H$ be a homogeneous manifold, where $G$ and $H$ are connected Lie groups and assume there is given a $G$-invariant Riemannian metric on $X$. Let $B(R)$ be the closed ball of radius $R>0$ ...
4
votes
1answer
110 views

The question about elementary equivalence of free products

Let $A,B,C,D$ be algebraic systems and $A$ and $B$ be elementary equivalent as well as $C$ and $D$. Are free products of $A,C$ and $B,D$ elementary equivalent if $A,B,C,D$ are groups, or $A,B,C,D$ ...
15
votes
3answers
964 views

Finitely generated matrix groups whose eigenvalues are all algebraic

Let $G$ be a finitely generated subgroup of $GL(n,\mathbb{C})$. Assume that there exists a number field $k$ (i.e. a finite extension of $\mathbb{Q}$) such that for all $g \in G$, the eigenvalues of $...
3
votes
1answer
59 views

Frattini subgroup is normal-monotone

On page 199 of Dummit and Foote's Abstract Algebra (Here $\Phi(G)$ is the Frattini subgroup of a group $G$, not necessarily finite): If $N\unlhd G$, then $\Phi(N)\subseteq\Phi(G)$. First, When ...