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

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4
votes
1answer
80 views

Scotts Theorem for one ended Fuchsian groups

in Peter Scott's work "Subgroups of Surface groups are almost geometric" is proven that for a closed surface $S$ and any fin. gen. subgroup $U$ of $G:=\pi_1(S,x)$ there exists a finitely sheeted ...
1
vote
1answer
62 views

A problem with pointwise stabilizer subgroups of fixed-point subspaces II

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$. Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$. Let ...
-3
votes
0answers
28 views

why the split extension of a quasicyclic $2$-group $C$ by the cyclic group is not finite by abelien [on hold]

Let $G$ be split extension of a quasicyclic $2$-group $C$ by the cyclic group of ordre 2 generated by the inversion automorphism of $C$ it is clear that $G$ is abelien by (finite cyclic) but why $G$ ...
4
votes
1answer
139 views

Represent matrix immanants using Schur functions

For each irreducible character $\chi^\lambda$ of the symmetric group $S_n$, the immanant of an $n\times n$ square matrix $A$ is defined as \begin{equation*} d_\lambda(A) := \sum_{\sigma \in S_n} ...
2
votes
0answers
112 views

One question about iteration on groups

Let $G$ be a finitely generated group, $H$ a subgroup of $G$ of index $n$, with $a_i$ a set of coset representatives and $$G=\displaystyle\bigcup_{i=1}^nH{a_i}.$$ Let $\phi:H\rightarrow G$ be a ...
0
votes
1answer
61 views

A problem with pointwise stabilizer subgroups of fixed-point subspaces I

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$. Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$. Let ...
23
votes
0answers
768 views

Why are there so few quaternionic representations of simple groups?

Having spent many hours looking through the Atlas of Finite Simple Groups while in Grad school, I recall being rather intrigued by the fact that among the sporadic groups, only one (McLaughlin as I ...
3
votes
0answers
55 views

Decidable properties of the Cayley complex of a presentation

Let $X= X(P)$ be the Cayley complex of a finite group presentation $P=<S | R>$. Are there geometric properties of $X$ that are known to be decidable by an algorithm that takes $P$ as input? For ...
10
votes
4answers
1k views

How many elements does it take to normally generate a group?

This is a terminology question (I should probably know this, but I don't). Given a group $G$, consider the minimal cardinality $nr(G)$ of a set $S \subset G$ such that $G$ is the normal closure of ...
3
votes
0answers
306 views

A dual version of a theorem of Øystein Ore in group theory

Let $(H \subset G)$ be an inclusion of finite groups. This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved: Theorem: $\mathcal{L}(H\subset G)$ ...
-2
votes
0answers
47 views

finite cyclic group have subgroup of prime index [on hold]

I'am trying to prove that if $G$ is finite cyclic group there is subroup of $G$ of prime index ,is it true?
4
votes
1answer
213 views

Is there a nonabelian finite simple group with Grothendieck ring of multiplicity one?

Let $G$ be a finite group. It admits finitely many irreducible complex representations $H_1, \dots, H_r$ which generate, for $\oplus$ and $\otimes$, the Grothendieck ring $\mathcal{G}(G)$ of $G$ (also ...
5
votes
4answers
658 views

Consequences of the Inverse Galois Problem

Are there any papers written about the consequences of the Inverse Galois Problem in case it is proved to be true or false? We know a lot of things that would be true if the Riemann Hypothesis holds. ...
0
votes
1answer
73 views

Unipotent orbit in adjoint group over finite field

[Editted: The assertion is wrong; see Jay's answer] My apology if this question is too simple. I am reading Deligne-Lusztig "Reductive groups over finite fields" and at the beginning of Chap. 4, ...
18
votes
6answers
1k views

Residual finiteness: why do we care?

Residually finite groups have been studied for a long time. However, I am struggling to work out why we care, or perhaps, why they continue to be of interest. Let me explain. Magnus, in his 1968 ...
3
votes
2answers
246 views

Adeles and twisted adeles

Let $\mu_n$ denote the group of $n$-th roots of unity in ${\mathbb{C}}$, i.e., $\mu_n=\ker[{\mathbb{C}}^*\overset{n}{\longrightarrow}{\mathbb{C}}^*]$. We set $$ \mu=\varinjlim_n \mu_n\subset ...
2
votes
1answer
154 views

Normal subgroup of a totally ordered group

A totally ordered group is a group equipped with a compatible total order, that is, $x\leq y$ and $z\leq t$ imply $x+z\leq y+t$ for all $x,y,z,t$ in the group. Is it true that every totally ordered ...
6
votes
1answer
151 views

Isometries of some simple Cayley graphs

Consider a Cayley graph of a group $G$ with respect to a symmetric finite generating set $S$. There are some obvious candidates to isometries of this graph - for example, translation by elements of ...
7
votes
1answer
208 views

Can you decide whether the commutator subgroup of a f.p. group is f.g?

Is the following algorithmic problem known to be decidable/undecidable? Input: a finite group presentation $P$. Decide: is the commutator subgroup of the group presented by $P$ finitely generated?
2
votes
1answer
153 views

Matrix Elements of Real Representations

I asked this question over at Math.StackExchange and despite having had a bounty on it I did not receive an answer. Suppose that $G$ is a finite group and we have a unitary irreducible representation ...
3
votes
0answers
148 views
+150

Alternating quotients of (2,3,7;10)

It was shown that the only quotient of the group $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{10} \rangle$ of the form PSL(2, q) is PSL(2, 41). That lead me to consider other families of simple ...
3
votes
1answer
106 views

Finite p-groups and their fibered products

Is every finite $p$-group an epimorphic image of a fibered product of two finite $p$-groups which can be generated by $2$ elements?
3
votes
1answer
713 views

Efficient algorithms to determine the roots of: $p(x) = r^x $ in the finite field $GF(q)$, where $r$ a primitive root of the field

I need to make sure that no efficient (i.e., polynomial time) algorithm exists for the following problem: Exponentiating Polynomial Root Problem (EPRP) Let $p(x)$ be a polynomial with $\deg(p) \geq ...
4
votes
1answer
121 views

Centralizers of reflections in special subgroups of Coxeter groups

Let us consider a (not necessarily finite) Coxeter group $W$ generated by a finite set of involutions $S=\{s_1,...,s_n\}$ subject (as usual) to the relations $(s_is_j)^{m_{i,j}}$ with ...
-3
votes
0answers
26 views

On finite lower central depth [closed]

Let G be group have finite lower central depth then evry subgroup of G must have finite lower central depth . i.e $\gamma _n(G)$=$\gamma _{n+1}(G)$ for some positive integer n we have to show ...
0
votes
0answers
33 views

induction on the nilpotency class [closed]

let G be group and N nilpotent subgroup of G such that $G/N$ is finite by nilpotent. denote $A$ the centre of $N$ . I wont to prove "By induction on the nilpotency class of $N$ we may asume that ...
5
votes
0answers
225 views

Does the Approximation Property (AP) pass to quotients by amenable subgroups?

Given a countable group $G$ with the AP, and an normal, amenable subgroup $N$ of $G$, does $G/N$ have the AP? In particular, does there exist a group $G$ with the AP and a surjective group ...
12
votes
2answers
364 views

generating set for symmetric group $S_n$

Say that $a_1, \cdots, a_{n-1}$ is an independent generating set for $S_n$. Let $b$ be any element in $S_n$. Is it true that $b$ can replace one of the generators, i.e. that there exists an index $i$, ...
21
votes
3answers
1k views

Are all free groups linear, i.e., admit a faithful representation to GL(n,K) for some field K ?

All free groups of finite or infinite countable rank are subgroups of the free non-abelian group $F_2$, which is linear. However, a free group of infinite uncountable rank will not be a subgroup of ...
4
votes
2answers
225 views

Maximal abelian subgroup of general linear groups

Thanks for any help or comments. Is it possible to recognize all maximal abelian subgroups of general linear group on finite field $F$ of order $q$, $GL_n(F)$. By maximal abelian I mean if $A$ is ...
-5
votes
0answers
38 views

elements of order two in SL2(R) [closed]

elements of order2 in Sl2(R) especilly in F( is filde) with odd character Whats elements of order 2.this is aproblem in linear groups I will find this elements of SL2(R)?
3
votes
1answer
124 views

About the second largest adjacency eigenvalue of Abelian Cayley graphs

[Assume all groups are finite] One knows the general statement that the sum of the values of the character function on the generating set is an eigenvalue of a Cayley graph. But the above doesn't ...
1
vote
0answers
77 views

project limit on $n$- simplical complex which is principal homogeneous with respect to an action

The setting: Let G be compact locally $\Bbb{Q}_p$ analytic group. We fix a countable basis of open normal subgroups $G\supset G_1\supset ...G_r\supset...$ We suppose that we are given a system of ...
5
votes
0answers
152 views

When do two lattices have the same stabilizer in the diagonal torus?

This is moved from MSE, where I asked and didn't receive an answer (see http://math.stackexchange.com/questions/1145151/lattices-in-mathbbq-pn-with-the-same-stabilizer) Let $T$ be the diagonal torus ...
2
votes
1answer
91 views

Is there a non right-orderable torsion-free factor of the Braid group on 3 strands?

The braid group on 3 strands has the presentation $\langle x,y \;|\; xyx=yxy\rangle$. A group $G$ is called right orderable if there is a total order $<$ on the set $G$ such that if $a<b$ then ...
3
votes
1answer
124 views

Subgroups of index 2 in a fibered product

Let $G$ be a finite group and let $M,N \lhd G$ be normal subgroups with a trivial intersection. Suppose that $G$ has a subgroup of index $2$. Must $G$ have a subgroup of index $2$ which contains ...
4
votes
1answer
118 views

Motivational ideas for the Gelfand-Graev character of a finite group of Lie type

I've been studying the Gelfand-Graev character's general construction for a finite group of Lie type. I wish to discuss its particularization in a seminar for the general linear group over a finite ...
4
votes
1answer
181 views

Quotient of principal congruence subgroups

This is a direct follow-up to this question. What is the quotient $\Gamma(2)/\Gamma(2^n)?$ (the principal congruence subgroups are in $SL(2, \mathbb{Z}).$ It is a 2-group, but what else?
3
votes
0answers
94 views

uniqueness of quotients of principal congruence subgroups

For each $n \geq 2$, is $\Gamma(2^{n})$ the unique normal subgroup of $\Gamma(2)$ with quotient isomorphic to $\Gamma(2) / \Gamma(2^{n})$ (here we are talking about principal congruence subgroups of ...
-1
votes
1answer
186 views
27
votes
2answers
1k views

How do you *state* the Classification of finite simple groups?

From the point of view of formal math, what would constitute an appropriate statement of the classification of finite simple groups? As I understand it, the classification enumerates 18 infinite ...
10
votes
2answers
584 views

Groups “approximately commutative” near the identity

Is the following idea something that is known? I call a metric group[1] $G$ "approximately commutative near the identity" if there exists a $K$ such that for small enough $\epsilon$, when $d(g,id) ...
7
votes
2answers
607 views

Translation length functions of non-simplicial trees

Let $G$ be a finitely generated group. By a theorem of Culler and Morgan, the set of non-abelian (not necessarily simplicial) minimal $\mathbb{R}$-trees with isometric $G$-action injects into the ...
4
votes
1answer
159 views

About the set of Sylow-$p$ subgroups of $G$

Let G be a finite group and S be the set of Sylow p-subgroups of G for a prime p dividing the order of G. Assume that |S|>1. Let U and V be two disjoint non-empty subsets of S such that, ...
1
vote
0answers
61 views

Counting group invariants using Macdonald conjecture?

It is known from Dyson, Macdonald $et$ $al$ that the constant term asscoiated to the expansion of the following expression \begin{equation} \prod_{\alpha \in \Delta} (1-e^{\alpha})^k \end{equation} ...
3
votes
0answers
170 views

every element with eigenvalue 1

Is there a "non-trivial" subgroup $G$ of $GL(n,\mathbb C)$ whose every element has an eigenvalue $1$? "Non-trivial" here means that is no common eigenvector with eigenvalue $1$ for all elements of ...
14
votes
1answer
586 views

In how many steps a random walk visits all the elements of a finite group, with a probability 1/2?

This question is a variation of the return to the origin problem. Let $G$ be the finite group $\mathbb{Z}/n \times \mathbb{Z}/n$ and let the random transformation $T: G \to G$ such that $T(a,b) = ...
6
votes
2answers
985 views

First group homology with general coefficients

When $G$ acts trivially on $M$, the first homology group is just the abelianisation of $G$ tensored with $M$, i.e. $H_1(G;M)=(G/[G,G])\otimes_\mathbb Z M$. Is there any similar statement when $G$ ...
3
votes
1answer
94 views

Section of Cayley graphs

Let $\pi: G\to G_1$ be a surjective group homomorphism to a finite group $G_1$ and let $S_1$ be a (finite) generating set of $G_1$. Assume $\mathrm{Cay}(G_1,S_1)$ is the Cayley graph of $G_1$ with ...
7
votes
3answers
324 views

Decision problem on triviality of intersection of two subgroups

What is known about the following decision problem? Given two finite sets in a finitely generated group G, decide whether the subgroups generated by them have trivial intersection. Is this problem ...