# Tagged Questions

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

**3**

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113 views

### Not too classical group characters

When teaching group theory, one has too speak of characters, which are morphisms $\chi:G\rightarrow k^\times$ into the multiplicative group of a field $k$ (mind that I don't mean representation ...

**2**

votes

**0**answers

41 views

### What is the minimal girth of a two-generator cayley graph for Alt(n) in which the girth relator is not a proper power?

First, a definition: a girth relator for a Cayley graph is a word that you get by reading the edge labels along a shortest loop. The girth relator is not usually unique, though it is often unique up ...

**2**

votes

**1**answer

168 views

### Is the centralizer $Z_G(A)=\{g\in G| a g= g a\}$ of a finite $A\subset G$ connected for a connected compact Lie group?

Let $G$ be a connected compact Lie group, consider the left/right action on itself.
For any finite $A\subset G$, consider the centralizer
$Z_G(A):=\{g\in G| a g= g a\}$.
Q: is $Z_G(A)$ a connected ...

**0**

votes

**0**answers

49 views

### Bases of surface groups with length restrictions

This question asks for a generalization of Bases of surface groups following the notation and definitions given therein.
Let $\Gamma_g$ be a surface group of genus $g \geq 2$, $B$ a surface basis of ...

**3**

votes

**0**answers

114 views

### What do we know about isospectral Cayley graphs?

Let $\Gamma_1=\Gamma_1(G_1,S_1)$ (respectively $\Gamma_2=\Gamma_2(G_2,S_2)$) be a Cayley graph for the group $G_1$ and the finite generating set $S_1$ (respectively $G_2$ and the finite generating set ...

**5**

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169 views

### A generalization of real characters on a group

Yesterday I understood that I can't live without this construction:
Let $G$ be a group, $A$ an associative algebra over $\mathbb R$ and $n\in{\mathbb N}$. We consider a sequence of maps ...

**2**

votes

**1**answer

129 views

### Is inner product preserved only by the stabiliser in a finite reflection group?

Is the following statement true for finite reflection groups?
Let $G$ be a finite reflection group acting on $\mathbb{R}^n$, let $x, y\in \mathbb{R}^n$
and let $z$ be in the orbit of $y$. If ...

**7**

votes

**2**answers

415 views

### Galois groups and prescribed ramification

What is known about finite groups $G$ for which there exists a Galois extension $K$ of $\mathbb{Q}$ ramified only at $2$ such that $\text{Gal}(K/\mathbb{Q}) \cong G$ ? More generally, which groups can ...

**2**

votes

**1**answer

120 views

### Subdirectly irreducible groups

A group is subdirectly irreducible provided it has a least nontrivial normal subgroup. Subdirectly irreducible groups are also referred to as monolithic groups in the literature. Every simple group is ...

**0**

votes

**1**answer

89 views

### On the realization of a compact surface as a leaf of an analytic foliation

Let $S$ be a compact orientable surface of genus $g \geq 2$. Is there any transversely real (or complex) analytic codimension one foliation $\mathcal{F}$ such that $\mathcal{F}$ has $S$ as a leaf with ...

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vote

**0**answers

100 views

### An epimorphism into a profinite group

Let $p$ be an odd prime number, $G$ a finitely generated nonabelian profinite group, $L \lhd_o G$ a pro-$p$ group with $[G : L] = 2$. Suppose that there is a continuous surjection from $G$ onto a free ...

**3**

votes

**1**answer

84 views

### A question about minimal nonnilpotent groups

Let $G$ be a minimal nonnilpotent group with cyclic Sylow $p$-subgroup $P$ and normal Sylow $q$-subgroup $Q$ [see Huppert, Endlich Gruppen I]. If $Q$ is abelian and $q > 2$, then can we get that ...

**0**

votes

**0**answers

38 views

### Bases for free pro-p groups

Let $p$ be a prime number, $F$ a free nonabelian finitely generated pro-$p$ group, $L \lhd_o F$ and $Y$ a basis for $L$ with $y \in Y$. Is there a basis $X$ for $F$ such that $y$ is in the abstract ...

**8**

votes

**1**answer

162 views

### Variety of nilpotent Lie algebras or $p$-groups

Here's a couple of analogous questions, one in terms of finite-dimensional complex Lie algebras and one in terms of finite $p$-groups; I'd be interested in an answer to either:
1) Let $\mathcal{L}$ ...

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votes

**0**answers

69 views

### Dense free subgroups

Let $F$ be a free pro-$p$ group (for a prime number $p$) on a finite set $X$, $\Phi$ the abstract subgroup generated by $X$, $\{1\} \neq N \lhd_c F$. Is it possible that $\Phi \cap N = \{1\}$?

**13**

votes

**2**answers

290 views

### Minimal number of generators for $GL(n,\mathbb{Z})$

$\DeclareMathOperator{\gl}{GL}\DeclareMathOperator{\sl}{SL}$From de la Harpe's book "Topics in Geometric Group Theory" I learnt that $\gl(n,\mathbb{Z})$ is generated by the matrices $$s_1 = ...

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vote

**0**answers

44 views

### Lower central series in a free pro-p group

Let $F$ be a nonabelian finitely generated free pro-$p$ group, $H \leq_c F$ of infinite index. Denote by $\{F_n\}_{n \in \mathbb{N}}$ the lower central series of $F$, and set $r_n = [F : F_nH]$.
Is ...

**0**

votes

**0**answers

65 views

### A bound on the size of the center

Let $p$ be a prime number, $F$ a free pro-$p$ group, $H \leq_c F$ of infinite index. Can it be that $$\sup_{N \lhd_o F} |Z((F/N)/C_{F/N}(HN/N))| < \infty ?$$

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161 views

### Finite Cohomology and free groups

Let $F$ be a finitely generated nonabelian free profinite group, $p$ a prime number, $L \lhd_o F$ with $[F : L]$ coprime to $p$, $N \lhd_c^\infty F$ contained in $L$ with $L/N$ pro-$p$, and $N \leq H ...

**4**

votes

**2**answers

179 views

### A Karrass-Solitar theorem for surface groups

Let $\Gamma_g$ be a surface group of genus $g \geq 2$. That is, there is a presentation $$\Gamma_g = \langle x_1, y_1, \dots, x_g, y_g \vert \prod_{i = 1}^{g}[x_i,y_i] = 1\rangle$$
Is there a ...

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92 views

### Golod Shafarevich Inequality and Inequalities among higher Cohomology groups

As a consequence of Golod- Shafarevich, we get an inequality between second cohomology group of a $p$-group with coefficients in $F_p$ and the first cohomology group of a $p$-group with coefficients ...

**5**

votes

**1**answer

168 views

### Bases of surface groups

Let $\Gamma_g$ be a surface group of genus $g \geq 2$. A $2g$-tuple $(x_1,y_1, \dots,x_g,y_g) \in \Gamma_g^{2g}$ will be called a Surface Basis if we have the presentation $$\Gamma_g = \langle x_1, ...

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**0**answers

88 views

### Periodic group with bound on order of finite subgroups

I have asked the same question previously on stackexchange without any answer (http://math.stackexchange.com/questions/923638/periodic-group-with-bounded-subgroups):
I am looking for infinite ...

**1**

vote

**0**answers

124 views

### Accessible subgroups of free groups

Let $F$ be a nonabelian free finitely generated group, and $F = G_0 \rhd G_1 \rhd G_2 \dots$ a strictly descending subnormal chain of subgroups ($G_n \lhd G_{n-1}$ for each $n \in \mathbb{N}$) each ...

**11**

votes

**1**answer

279 views

### Union of conjugates in free groups

Let $F$ be a (finitely generated) free group, $H \leq F$ of infinite index. Is it possible that $$ \bigcup_{g \in F} gHg^{-1} = F?$$

**0**

votes

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61 views

### A normal form theorem for presentations of finite $p$-groups of nilpotency class $2$?

When constructing examples of nonabelian finite $p$-groups with abelian automorphism group (and certain other desired properties), the authors of papers like http://arxiv.org/pdf/1304.1974v1.pdf leave ...

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68 views

### Confusion about the projected component in an irreducible space in the tensor product decomposition using Littlewood-Richardson?

The regular representation of the symmetric group can be formulated in terms of an abstract tensor, where the action of the symmetric group elements is by means of permuting the indices. Given an ...

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votes

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420 views

+150

### Distributivity of group topologies on $\Bbb Z$

Let $\mathcal L$ be the set of all group topologies on $\Bbb Z$.
It is known that $(\mathcal L,\subseteq)$ is a modular complete lattice [1].
Is $(\mathcal L,\subseteq)$ distributive?
$$~$$
...

**10**

votes

**2**answers

489 views

### (co)homology of symmetric groups

Let $S_n=\{\text{bijections }[n]\to[n]\}$ be the n-th symmetric group. Its (co)homology will be understood with trivial action. What are the $\mathbb{Z}$-modules $H_k(S_n;\mathbb{Z})$? Using GAP, we ...

**4**

votes

**1**answer

173 views

### Ends of Coxeter Groups

It is known after Stallings that a group can have 0, 1, 2 or infinitely many ends. Are there known results on the space of ends of a Coxeter group?

**11**

votes

**1**answer

529 views

### In what sense is the classification of all finite groups “impossible”?

I think there is a general belief that the classification of all finite groups is "impossible". I would like to know if this claim can be made more precise in any way. For instance, if there is a ...

**2**

votes

**2**answers

140 views

### Quasi-isometry and left invariant orderability for groups

Is the property of left invariant orderability for finitely generated groups preserved by quasi-isometrics? More precisely, if $G$ is a left orderable (finitely generated) group and $H$ is a ...

**6**

votes

**1**answer

237 views

### An inequality on representations and subgroups of general linear groups over finite field

Let $q$ be a power of $p$, let $l$ be a prime different from $p$, and let $H_1$ and $H_2$ be two subgroups of $GL_n(\mathbb F_q)$ that are $l$-groups.
If for all characteristic $0$ representations ...

**11**

votes

**1**answer

274 views

### What are smallest finite images of triangle groups?

Given integers $a, b, c$ all at least 2, I would like to identify the smallest group with an element $x$ of order $a$, element $y$ of order $b$ such that the product $yx$ has order $c$. For example, ...

**7**

votes

**1**answer

201 views

### Percentage of elements that can be written as commutators

Let $G$ be a finite perfect group. Are there any results on the number of elements of $G$ which can be written as a commutator? When $G$ is finite non-abelian simple group, then every element can be ...

**0**

votes

**0**answers

43 views

### Is the projector to irreducible tensor modules of SO(N) known?

To project a generic tensor to an irreducible module of SO(N) one has to (anti)symmetrize the indices and then subtract traces, e.g. for symmetric traceless 2-tensors
$$
\frac{1}{2} (\delta_{I_1 J_1} ...

**10**

votes

**2**answers

262 views

### existence of a finite group which is the union of self normalizing subgroups

Can a finite group G be the union of self normalizing subgroups such that the intersection between any two of these subgroups is equal to the unit of the group G? I don't think so but I can't prove ...

**0**

votes

**0**answers

98 views

### Soluble group algebras and centralizers

Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and every ...

**8**

votes

**2**answers

274 views

### Sums of degrees of irreducible complex characters

The sum of the degrees of the irreducible complex characters (not the square sum which is the group order) is relevant to determine the dimension of a maximal torus of the Lie algebra associated to ...

**5**

votes

**0**answers

143 views

### (Connected) Cayley graphs of PSL(2,q) from (2,3,n)-triples

Let $G = PSL(2,q)$. I'm interested in the Cayley graphs of $G$ generated by triples $(A,BAB^{-1},B^{-1}AB)$, where $A, B \in G$ are elements of order $2, 3$ respectively: such a triple generates all ...

**10**

votes

**1**answer

257 views

### Which finite simple groups can be characterized by their action on a small set?

It is well known that a finite 4-times transitive permutation group is Matthieu, symmetric, or alternating. Another way of stating this is that the set
$$
\Omega = \{(x_1, x_2, x_3, x_4), 1\leq ...

**1**

vote

**0**answers

47 views

### on reductive monoids which are gorenstein

Let $M$ a reductive monoid, i.e. a integral normal affine scheme, which is a monoid whose group of units is a connected reductive group.
By Rittatore ...

**7**

votes

**1**answer

404 views

### Group structure on an arbitrary completely regular topological space that makes $(x,y)\mapsto xy^{-1}$ continuous at $(1,1)$

Let $(G,\mathcal T)$ be a completely regular topological space. Is there a group structure on $G$ such that the function
$$f:G\times G\to G$$
$$f(x,y)=xy^{-1}$$
is continuous at $(1,1)$?

**7**

votes

**4**answers

625 views

### Ore's Conjecture for perfect groups

We know that Ore's Conjecture is a theorem now. Does anyone know a counterexample to Ore's conjecture for perfect groups? I believe there should be one, but I dont have a counterexample. Thanks.

**2**

votes

**1**answer

170 views

### Quotient of Projective line over rationals with an infinite subgroup of PGL(2,Q)

I am looking for references for the following; how to calculate quotient of the projective line over the field of rationals with an infinite subgroup of PGL(2,Q), e.g, of the form
$
\left(
...

**2**

votes

**1**answer

146 views

### Classification of indecomposable inclusions $(H \subset G)$ with $G$ decomposable

Definition: A group $G$ is indecomposable if: $G = G_1 \times G_2 \Rightarrow \exists i \ G_i = 1$.
We can generalize the notion of indecomposable from groups to inclusion of groups as ...

**6**

votes

**2**answers

299 views

### Do compact groups acting irreducibly have finite subgroups which do the same?

Let $G$ be a closed subgroup of $U(n,{\bf C})$, not necessarily connected. Regard ${\bf C}^n$ as a complex $G$-module $M$.
Q. Suppose $M$ is irreducible as a $G$-module (equivalent, I think, to ...

**7**

votes

**1**answer

426 views

### Generalization of a theorem of Øystein Ore in group theory

Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive.
Proof: see below.
Let $(H \subset G)$ be an inclusion of finite groups and ...

**2**

votes

**0**answers

235 views

### Prime order elements in $GL(n,\mathbb{Z})$

What is known about the elements of prime order $p$ in $GL(n,\mathbb{Z})$? All I know at this point is that $GL(n,\mathbb{Z})$ has an element of prime order $p$ iff $p-1\leq n$ and that there are only ...

**1**

vote

**2**answers

165 views

### Cubic Cayley (undirected) graphs

The generators of a cubic Cayley graph always include at least one involution, since 3 is odd. There are then two possibilties for the other two generators: either (A) they are also (distinct) ...