# Questions tagged [gr.group-theory]

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

5,535 questions

**10**

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97 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**

vote

**0**answers

59 views

### 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\...

**6**

votes

**0**answers

216 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

**2**answers

188 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

**0**answers

79 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 ...

**1**

vote

**2**answers

216 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

**1**answer

265 views

### Variation on Sylow Theorems [on hold]

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 ...

**2**

votes

**1**answer

130 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

**0**answers

98 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

**0**answers

88 views

### Formal sums on groups with symmetric finite presentations [on hold]

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

**0**answers

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

**0**answers

39 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

**0**answers

50 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
$G$...

**18**

votes

**0**answers

309 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

**1**answer

103 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

**0**answers

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

**0**answers

73 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

**1**answer

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

**0**answers

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

**0**answers

89 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

**1**answer

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

**1**answer

82 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

**0**answers

60 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

**0**answers

143 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

**0**answers

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

**1**answer

155 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

**1**answer

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

**1**answer

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

**1**answer

150 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 ...

**0**

votes

**0**answers

248 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

**1**answer

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

**0**answers

56 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

**0**answers

101 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

**0**answers

63 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

**0**answers

133 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

**2**answers

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

**1**answer

172 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

**1**answer

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

**1**answer

108 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

**1**answer

131 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

**0**answers

119 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

**1**answer

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

**1**answer

148 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

**1**answer

107 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

**3**answers

955 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

**1**answer

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 ...

**1**

vote

**1**answer

169 views

### Product of two group morphisms not a group morphism

In Mac Lane's Categories for the Working Mathematician, page 110, second edition, he states that, in the category of groups $Grp$, $F_n$ being groups and $x_n, y_n$ being two cones on these groups (...

**6**

votes

**1**answer

215 views

### Is a finite group given by its character table if its Sylow subgroups are so?

As pointed out by Mikko Korhonen in this answer, Özdem Çelik proved (in 1976 here) that a finite group whose Sylow subgroups are cyclic (called a Z-group) is determined by its character table.
...

**4**

votes

**0**answers

61 views

### What are the zonal spherical functions for a finite unitary group acting on a unit sphere?

Given a prime power $q$ and a dimension $d$, consider the Hermitian form $(\cdot,\cdot) \colon \mathbb{F}_{q^2}^d \times \mathbb{F}_{q^2}^d \to \mathbb{F}_{q^2}$ given by
$$
(x,y) = \sum_{i\in [d]} ...

**4**

votes

**0**answers

66 views

### Hilbert space compression of lamplighter over lamplighter groups

$C_2 \wr \mathbb{Z}$ is the lamplighter group but I'm currently looking at the lamplighter group with this group as a base space.
Question: Consider the group $C_2 \wr (C_2 \wr \mathbb{Z})$, what is ...