# Questions tagged [gr.group-theory]

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

5,763
questions

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### Number of commuting pairs in p-group

Let $p$ be a prime number and $G=GL_n ( \mathbb{Z} / p \mathbb{Z}
)$. Consider the set $U$ of upper-triangular matrices of $G$ having
entries of $1$ on the diagonal. The set $U$ is a Sylow $p$-...

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22 views

### Does Levi operator always map one-word varieties to one-word varieties?

Suppose $\mathfrak{U}$ is a group variety. Now let’s define $L(\mathfrak{U})$ as the class of all such groups $G$, such that $\forall g \in G$ $\langle \langle g \rangle \rangle \in \mathfrak{U}$ (...

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87 views

### Which irreducible representations of the symmetric group are eigenspaces of class sums?

In the setting of complex representations of finite groups, a class sum $1_C=\sum_{g\in C} g$ acts on an irreducible representation $V$ as $\lambda(C,V)\operatorname{Id}$, where $\lambda(C,V)=|C|\...

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105 views

### For a finite subgroup of an infinite group $G$, does its normal closure have infinite index in $G$?

For a finite subgroup of an infinite group $G$, does its normal closure have infinite index in $G$?
Or if not, is it true when we replace $G$ by some subgroup? That is:
Let $H$ be a finite ...

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**2**answers

161 views

### Mackey theory in the setting of locally profinite groups

$\DeclareMathOperator\Hom{Hom}$Let $R$ be a commutative ring (not necessarily unital). Let $G$ be a finite group, and let $H_1, H_2$ be subgroups of $G$.
Recall the following standard result [1, Thm. ...

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93 views

### Is there exists a lattice isomorphism?

Let $\text{P}$ be the set of partitions of {1,2,...,n} and $\text{Y}$ the set of Young subsets of permutation group S(n)(the coxeter group of type An).
As is well-known, the set $\text{Y}$ is a ...

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119 views

### Steinberg relations for elementary subgroup of a Chevalley group over an arbitrary ring

Given a semisimple Lie algebra $\frak{g}$ of type $\Phi$ with a Lie algebra representation $\rho:\frak{g}\to \frak{gl}(v)$ and an arbitrary commutative ring one can associate the following gadgets:
...

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**1**answer

137 views

### Uniform Roe algebra of virtually abelian group is type I C*-algebra?

Let $G$ be an arbitrary (discrete) group. It acts by left translation on $\ell^\infty(G)$. The uniform Roe algebra of $G$ is defined as the crossed product $\ell^\infty (G) \rtimes_{\mathrm{red}}G$.
...

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109 views

### $G$- space is locally compact [on hold]

Suppose $X$ is a topological space ,$G$ Is a locally compact group.If the quotient space $G\backslash X$ is compact,can we deduce that $X$ is locally compact?

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26 views

### Do all maximal verbal series have the same length?

Suppose, $G$ is a finite group. Let’s call a series of subgroups of $G$ $\{H_k\}_{k=1}^n$ a verbal series of length $n$, iff $H_1 = E$, $H_n = G$ and $\forall k < n$ $H_k$ is a verbal subgroup of $...

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145 views

+50

### Subset of reals associated to pairs of matrices in $\mathrm{SL}(2,\mathbb{R})$

Let $\Gamma$ be a subgroup of $\mathrm{SL}(2,\mathbb{R})$. I would like to ask if there is any research on the following set:
$$\Gamma*\Gamma:=\bigg\{\dfrac{(a+b)(a'+b')}{(c+d)(c'+d')}\bigg|\begin{...

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179 views

### Universal group on $\kappa$ elements

It is well known that for every positive cardinal $\kappa$, every group of cardinality $\kappa$ can be embedded into $\text{Sym}(\kappa)$, the group of bijections on $\kappa$ with composition as group ...

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**1**answer

288 views

### Gromov hyperbolic groups which are solvable are elementary

I have read on wikipedia that a Gromov hyperbolic group which is solvable is elementary (i.e. virtually cyclic). Where can I find a proof of this fact?
There is a proof of a similar fact in Bridson-...

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68 views

### Finite groups of cyclicality index $3$

Suppose $G$ is a group. Let’s define the cyclicality index of $G$ using the following recurrent relation:
$$CI(G) = \begin{cases} 1 & \quad G \text{ is cyclic} \\ \max_{H < G} CI(H) + 1 & \...

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58 views

+50

### Efficiently computable graph conductance measures

Treating electrical networks from a graph theory point of view, do there exist measures that characterize the overall electrical conductance of the graph whilst being efficiently computable?
There ...

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**2**answers

322 views

### Does asymmetric fraction of finite groups tend to $0$?

Let’s define asymmetric fraction of a finite group $G$ as the number $$\mathrm{af}(G) = \frac{|\{(g, a) \in G \times \mathrm{Aut}(G)\mid a(g) = g\}|}{|G|\cdot|\mathrm{Aut}(G)|}.$$ Equivalently it can ...

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63 views

### Amalgamated subgroup of an HNN extension finitely generated

Baumslag proved that if $G= A \ast_{C} B$ is an amalgamated free product where $A$ and $B$ are finitely presented, $G$ is finitely presented if and only if $C$ is finitely generated.
Similarly, by ...

**6**

votes

**1**answer

258 views

### Wreath product $S_k\wr S_n$ inside $S_{kn}$

I want to understand wreath products a little better.
Currently, my intuition about them is as follows. Take $nk$ disks, pile them in order forming $n$ piles of $k$ disks (one pile contains disks {1,...

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131 views

### Writing canonically a transitive action as quotient of a simply transitive action

Consider a finite group $G$ acting transitively on a finite set $Y$. Is it possible to find a finite set $P_Y$ and a finite group $\hat G$ acting on $P_Y$ such that the following hold?
The action of $...

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**1**answer

78 views

### Union of an ascending chain of subgroups in group $G$ isomorphic to subgroup $S_0\subseteq G$ [closed]

Let $G$ be an infinite group, let $S_0\subseteq G$ be a subgroup and suppose that ${\frak C}$ is a collection of subgroups of $G$ such that
$C \cong S_0$ for all $C\in {\frak C}$, and
for all $C, C'\...

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161 views

### Algebraic varieties associated to finite groups

Have the following equations been studied in the literature?
Let $G$ be a finite group.
Then I am looking for functions $f : G \rightarrow \mathbb{C}~ \backslash \left\lbrace 0 \right\rbrace $ such ...

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votes

**1**answer

55 views

### Is $S_\omega/F_\omega$ embeddable to $S_\omega$?

Let $S_\omega$ be the group of bijections $f:\omega\to\omega$, and let $F_\omega = \{\pi\in S_\omega: \exists N\in \omega(\pi(k) = k \text{ for all } k\geq N)\}$. It is easy to see that $F_\omega$ is ...

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287 views

### Can the set of endomorphisms of $(\mathbb R,+)$ have cardinality strictly between $\frak c$ and $\frak{c^c}$?

Let $\frak c$ be the cardinality of the reals. I know that in ZF the set of endomorphisms of $(\mathbb R,+)$ can have at least two different cardinalites:
If we allow the axiom of choice, you can ...

**20**

votes

**3**answers

2k views

### In what sense is SL(2,q) “very far from abelian”?

I am far from an expert in this area.
I'd be grateful if someone could explain in what sense $\mathop{SL}(2,q)$ is
"very far from abelian," to quote
Emanuele Viola?
Why does Theorem 1 (below) justify ...

**0**

votes

**1**answer

116 views

### Number of cycles under a certain action on Z/nZ [closed]

Computer scientist here looking at a question that came about from in-place matrix transposition, but rusty on my abstract algebra and number theory...
Suppose we have the multiplicative group $\...

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89 views

### On 2 crossed modules

Let $(G,H,\alpha,\tau)$ and $(H,J,\alpha',\tau')$ be two crossed modules of groups. It is given that $Kernel(\tau) \cap Image(\tau')$ =$e$. For every $h\in H$ does there always exist a $j_h \in J$ ...

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152 views

### How bad can the orbit space (assume it‘s finite) be?

Fix a connected algebraic agroup $G$ over a local field $F$ (e.g real numbers). Consider all varieties $X$ over $F$ with $G$ action such that $O_X=X(F)/G(F)$ is finite. Using anayltic topology $X(F)$ ...

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138 views

### Finite index subgroup of HNN extension

Let $GX$ be a tree group (a right-angled Artin group such that the graph is a tree), such that not all the subgroups of $GX$ are necessarily RAAGs, so the length of the tree is greater or equal than $...

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86 views

### When are extensions of algebraically good groups algebraically good?

Let $G$ be a discrete group. The pro-algebraic completion of $G$ is a pro-algebraic group $G^{\mathrm{alg}}$ together with a morphism $s:G\to G^{\mathrm{alg}}$ which is initial among all morphisms ...

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69 views

### Spacetime symmetries

We know some nice space-time have a lot of symmetries. It is said that
Minkowski spacetime has
$$ISO(d-1,1)/SO(d-1,1),$$
de Sitter spacetime has
$$SO(d,1)/SO(d-1,1)$$ and
anti-de Sitter spacetime ...

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votes

**1**answer

145 views

### Anti-holomorphic involutions of a complex linear algebraic group

Let $G$ be a connected linear algebraic group over the field of complex number ${\Bbb C}$.
Let $G({\Bbb C})$ denote the complex Lie group of ${\Bbb C}$-points of $G$.
Let $\sigma$ be an anti-...

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**1**answer

113 views

### Commensurator of a subgroup of matrices

Let $k$ be a totally real number field and let $\mathcal{O}_k$ denote its ring of integers. If $H$ is a subgroup of $\text{GL}(n, \mathbb{R})$ let denote with $H(k)$ and $H(\mathcal{O}_k)$ the ...

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96 views

### Is there an equivariant simplicial deformation retract of Teichmüller space?

Let $S_g$ be a surface of genus $g \ge 2$. By analogy with Teichmüller space for $S_g$, Culler and Vogtmann studied Outer Space $CV_n$, with points projective classes of marked metric graphs with ...

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91 views

### Generating the monoid of injections of the free group into itself

Let $F$ be the free group of rank $2$ (or any finite rank if this does not matter). The set of injections $F\to F$ forms a monoid $M$ by compositions. Is there a simple looking set of generators of $M$...

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**1**answer

117 views

### Diagonal automorphisms for twisted Chevalley groups

Let $G$ be a Chevalley group over a field $k$ of characteristic $0$. We know that a diagonal automorphism $\phi_h$ of $G$ is of the form $g\mapsto hgh^{-1}$, where $h\in \hat H$ and $\hat H$ ...

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73 views

### On normalized 2-cocycle

Let $G$ be a group acts trivially on an abelian group $A$. Let
$\varepsilon $ be a normalized 2-cocycle in $ Z^{2}(G,A)$. Assume
that $G=H_{1} \times H_{2}$ and let $\varepsilon_{1}=res_{H_{1}\times
...

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132 views

### Existence of n-axial elements in groups with at least 2 ends

Let $G$ be a finitely generated group. Fix some symmetric finite generating set $S$ for $G$, and write $\Gamma$ for the Cayley graph of $G$ with respect to $S$.
Given finite subsets $X,S,Y$ of $G$, ...

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295 views

### Is random walk drift rational?

(Question mildly edited for clarity by request of Matt F.)
If $G$ is a finitely presented group, let $|\cdot|$ denote the word metric with respect to a finite set of generators. Suppose $\nu$ is a ...

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164 views

### Diameter of finite rational matrix groups

Suppose $G$ is a finite subgroup of $\mathrm{GL}(n,\mathbb{Q})$.
For a set $\mathcal{M} \subseteq G$ that generates $G$, define the $\mathcal{M}$-diameter $\mathit{diam}(G, \mathcal{M})$ of $G$ to be ...

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**1**answer

142 views

### Cohomology of linear algebraic groups

Let $R$ be a commutative ring. Let $G\subset \mathrm{GL}_m$ be a linear algebraic subgroup. Has the group cohomology $H^i(G(R),R^m)$ been studied in the literature?
For example, do we know
(1) $H^...

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**1**answer

89 views

### Primitive action of wreath product

I say in advance that I am really new to Group Theory, so if my question is trivial I apologize in advance.
Let $A$ and $H$ be groups and $\Omega$ be a $H$-set. In this set-up, we can define the ...

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83 views

### Change variable in integration with symmetry

Not sure if I can ask such fundamental problem here.
Let
$G$ be a finite group, $\sigma \in G$. Consider linear group actions fo $G$ on $\mathbb{R}^n$.
$\sigma(K)=\{x\in \mathbb{R}^n: \sigma^{-1}(...

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109 views

### Description of quasimorphisms of the free group

Let $F$ be a free group of finite rank with a fixed basis and corresponding word metric. Let $Q = Q^0_h(F, \mathbb{R})$ be the space of real homogenous quasimorphisms that vanish on the basis of $F$. ...

**5**

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**1**answer

186 views

### Relations between boundaries of groups acting on hyperbolic spaces with WPD elements

Let $(X,d)$ be Gromov-hyperbolic space and let $\Gamma$ be a finitely generated group acting on $\Gamma$ by isometries. Recall the following two definitions.
Say that the action is acylindrical if ...

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66 views

### How are reflection groups related to general point groups?

I always tried to understand how the finite reflection groups of $\Bbb R^d$ (of some fixed dimension $d$) relate to the point groups of the same space $\smash{\Bbb R^d}$ (finite subgroup of the ...

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50 views

### Number of conjugacy classes of unit groups of modular group algebras

Let $n$ be a natural number, $p$ a prime number, $G$ a finite $p$-group and $K$ a finite field with $p^n$ elements. We focus on the group $1+J(KG)$, where $J(KG)$ is the Jacobson radical of $KG$, ...

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**1**answer

125 views

### Is it possible to extend this homomorphism?

Let $G$ be a torsion free group and $\alpha$ be a non-zero element in its complex group algebra. Assume that $\mathfrak A$ is the Banach sub-algebra of $\ell^1(G)$ generated by $\alpha$. Is it ...

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84 views

### Do positive-density subgroups intersect nontrivially?

Let $G$ be an infinite finitely generated group and $S$ a generating set. Define density with respect to the sequence of balls $S^n$.
If $H_1, H_2 \leq G$ have positive density, must $H_1 \cap H_2$ ...

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188 views

### Number of labeled Abelian groups of order n [migrated]

I calculated the number of labeled Abelian groups of order $N$ (i.e., the number of distinct, abelian group laws on a set of $N$ elements). This sequence is given by OEIS A034382, but my solution ...

**13**

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**1**answer

381 views

### Dimensional gap of group representations

The problem is inspired by eigenvalue bounds of random Cayley graphs on $SL_2(q)$.
Definition. An infinite series of finite groups $S$ is α-rich if the dimension of the smallest nontrivial ...