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

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-2
votes
0answers
44 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?
3
votes
0answers
52 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 ...
4
votes
1answer
206 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 ...
3
votes
1answer
119 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
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 ...
0
votes
1answer
72 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, ...
3
votes
2answers
242 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 ...
6
votes
1answer
149 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
203 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
1answer
104 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
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 ...
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 $i$, ...
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 ...
-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)?
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 ...
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
76 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 ...
3
votes
1answer
123 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 ...
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 ...
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?
5
votes
4answers
655 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. ...
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
185 views

Suppose that $G$ is a subgroup of $GL_n(\mathbb C)$ with finite exponent. Then is $G$ a finite group? [closed]

As title. the exponent of $G$ is the least number $n$ (if exists) such that $g^n=e$ holds for all $g\in G$ or $+\infty$.
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 ...
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 ...
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
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 ...
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
582 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) = ...
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 ...
3
votes
1answer
170 views

Conjugation of group extensions

Let $H$ be a finite group. We write ${{\mathbb{C}}}^{*n}$ for the $n$-dimensional complex torus $({{\mathbb{C}}}^*)^n$. We have a short exact sequence $$ 0\to {{\mathbb{Z}}}^n\to ...
9
votes
1answer
322 views

History of Tarski's problems on free groups

As is known, Tarski posed his questions about first-order theories of non-abelian free groups around 1945. However, the questions were not published in his papers or books. What is the original ...
5
votes
1answer
266 views

When are (Abelian) Cayley graphs also expanders?

I want to ask the question in two parts, (1) Is there some fundamental distinguishing property between Abelian and non-Abelian Cayley graphs? (say some specific proof technique which distinguishes ...
5
votes
0answers
223 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 ...
4
votes
0answers
149 views

Infinite simple p-groups with only trivial irreps in characteristic p

Is there a prime $p$ and an infinite simple $p$-group $G$ such that for any field $K$ of characteristic $p$ the only irreducible $KG$-module, whether finite or infinite dimensional, is trivial (that ...
3
votes
1answer
195 views

For a cross section $\sigma\colon G/N\to G$, how is $\sigma(y)^{-1}\sigma(x)^{-1}\sigma(xy)$ called?

Let $G$ be a locally compact group, let $N$ be a closed normal subgroup of $G$, and let $\sigma\colon G/N\to G$ be a cross section. Let us define $\alpha\colon G/N\times G/N \to N$ by the formula $$ ...
4
votes
1answer
268 views

Frequency of a representation of SO(3)

When generalizing the basic tenets of Fourier Theory to the symmetric group $S_n$, we can define a notion of the frequency of a basis function (i.e. an irreducible representation of $S_n$). In ...
4
votes
1answer
207 views

Decomposing $(\mathbb C^n)^{\otimes m}$ as a representation of $S_n\times S_m$

$V=\mathbb C^n$ is a $\mathbb CS_n$-module, where $S_n$ is the symmetric group of degree $n$, via the representation sending a permutation to the corresponding permutation matrix. The tensor power ...
3
votes
0answers
214 views

Galois correspondence subgroups/subsystems

In this paper (1998) by M. Izumi, R. Longo, S. Popa, there is the following result (page 49) on compact groups: By applying this result to finite groups, we get a Galois correspondence ...
1
vote
0answers
120 views

Does SL(3,q) have a subgroup of order $q^3.(q^3-1)$ [closed]

Let $q=p^n$ for $p>3$.I want to know whether the group $G_2(q)$ has a subgroup of order $q^3.(q^3-1)$. First I look the paper of Peter Kleidman. In this paper Kleidman determined all maximal ...
2
votes
1answer
154 views

What is the growth of the rank of a power of a finite simple group?

Which asymptotic bounds (upper and lower) are known for $s_n$ - the minimal number of generators of $S^n$ where $S$ is a nonabelian finite simple group?
10
votes
0answers
200 views

Solving a set of equations in a finite symmetric group

A standard way to find solutions to a finite set of equations in a finite symmetric group ${\rm S}_n$ is to take the equations as relators of a finitely presented group, to use the low index subgroups ...
2
votes
1answer
154 views

About direct limit of groups

Let $G_i$ be sequence of groups for $i\in \mathbb N$ and Let $\phi_i$ be a monomorphism from $G_i$ to $G_{i+1}$. Let $\Sigma$ be the direcet limits of $G_i$ under the embeddings of $\phi_i$. Let ...
3
votes
0answers
139 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 ...
4
votes
1answer
92 views

Finiteness properties for graph of groups decompositions

My curiosity was raised by the following question and the huge variety of comments and suggestions it attracted. I wondered if a converse statement might be equally interesting. Let $G$ be a finitely ...
3
votes
2answers
247 views

If d(“G/H”) < d(G) = 2, must H contain a primitive element?

Let $G$ be a finite group that can be generated by $2$ elements, and let $H \leq G$ be a (not necessarily normal) subgroup for which there exists some $g \in G$ such that $H \langle g\rangle = G$. ...
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 ...