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

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3
votes
1answer
175 views

Weyl groups of $E_6$ and $E_7$

The Weyl group $W_6$ of the Lie algebra $E_6$ is of order 51840, the automorphism group of the unique simple group of order 25920, while the Weyl group $W_7$ of the Lie algebra $E_7$ is of order ...
7
votes
1answer
301 views

Seeking conceptual explanation of these nice bijections on roots of unity

I proved the following facts by unenlightening calculations. Since the statements are quite clean, I think there should be a conceptual explanation for them, which my proof certainly is not. Let $q$ ...
2
votes
0answers
104 views

Over which fields is a $G$-module reducible?

I have asked this question at math.stackexchange, but have not received an answer so far. Also, I'm not entirely sure that this is a suitable question for mathoverflow... See this link for the ...
5
votes
0answers
56 views

Centralizers in virtually special groups

Let $G$ be a virtually compact special group, in the terminology of Haglund and Wise (i.e., $G$ has a finite index subgroup $H$ which is isomorphic to the fundamental group of some compact special ...
1
vote
1answer
76 views

Generating subgroups of large index by a large chunk of a conjugacy class

Let $G$ be a finite simple group and let $C$ be a (non-trivial) conjugacy class of $G$. Let $H$ be a subgroup of $G$ such that $$|H\cap C| \geq \epsilon |C|.$$ Can one conclude that the index of $H$ ...
17
votes
2answers
916 views

Is a group uniquely determined by the sets $\{ab,ba\}$ for each pair of elements a and b?

This is a cross-posted question, originally active here on math.stackexchange. For a given group $G=(S,\cdot)$ with underlying set $S$, consider the function $$ F_G:S\times ...
1
vote
1answer
142 views

Commutator with a generator of a free group

Let $F$ be a free group $\langle x_1,...x_m\rangle$. If $a\in F_2$ and $[a,x_1] \in F_n$ then $a\in F_{n-1}$. Here, $F_n$ is the $n$-th lower central series term with $F_2=[F:F]$. How can I prove ...
3
votes
2answers
147 views

About the number of their conjugacy classes in some classes of finite simple groups

We know that the orders of simple groups $B_n(q)$ and $C_n(q)$ are equal. What about the number of their conjugacy classes? Are they equal or not? Any reply, comment, remark or reference is ...
2
votes
1answer
828 views

Automorphism theorem

Help me please to find reference for the proof of the following theorem: Theorem. Let $\theta$ be a Leibniz cocycle on the Leibniz algebra L with values in $V,$ and assume $\theta^{\bot}\cap ...
3
votes
0answers
107 views

Growth of the number of generators in hyperbolic groups

Let $G$ be an infinite hyperbolic group, and let us further assume that it is residually finite (or even LERF/GFERF) so that we have plenty of subgroups of finite index. I would like to know if one ...
50
votes
6answers
10k views

Is Thompson's Group F amenable?

Last year a paper on the arXiv (Akhmedov) claimed that Thompson's group $F$ is not amenable, while another paper, published in the journal "Infinite dimensional analysis, quantum probability, and ...
3
votes
0answers
133 views

center of centralizer in finite group

My general question: Is there any reference for the center of centralizer in finte group. In particular for the element $x\in G$ such that $Z(C_G(x))=\langle x\rangle$. My motivation: Espacially when ...
0
votes
0answers
73 views

Carlson's translatability

I asked the following on MSE a few weeks ago but I did not get any answer : http://math.stackexchange.com/questions/1039593/carlsons-translatability-are-theses-characterisations-equivalent Given a ...
6
votes
1answer
217 views

what is the intersection of all congruence subgroups of the profinite completion of SL(2,Z)?

Let $\widehat{SL(2,\mathbb{Z})}$ be the profinite completion of $SL(2,\mathbb{Z})$. Let $\Gamma(N)$ denote the typical principal congruence subgroup of $SL(2,\mathbb{Z})$ (ie, all matrices congruent ...
0
votes
1answer
59 views

Powers in compact coset spaces

Let $G$ be a topological group, let $K$ be a closed cocompact subgroup (i.e. the coset space $G/K$ is compact in the quotient topology) and let $g \in G$. Is there a sequence (edit: or net) of ...
2
votes
1answer
110 views

Measuring products of finitely generated subgroups of free groups

Let $F$ be a finitely generated free group, $H_1, \dots, H_n$ finitely generated subgroups of infinite index in $F$, and $\epsilon > 0$. Must there be an epimorphism to a finite group $\phi \colon ...
8
votes
2answers
266 views

Can a positive measure subset of a free group be nowhere dense?

Let $F$ be a finitely generated free group and let $S \subseteq F$ be a subset for which there is some $\epsilon > 0$ such that for any epimorphism to a finite group $\phi \colon F \to G$ we have ...
5
votes
1answer
223 views

Is there a “good” reason that the universal central extension of $SL(2,\mathbb Z)$ is $Br_3$?

Let $$ Br_3 := \langle \tau_1,\tau_2\ :\ \tau_1 \tau_2 \tau_1 = \tau_2 \tau_1 \tau_2 \rangle $$ be the braid group on three strands, and consider the surjection $$\phi : Br_3 \twoheadrightarrow ...
2
votes
1answer
588 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 ...
0
votes
1answer
121 views

Murray–von Neumann equivalence on C$^*$-algebra and von Neumann algebra

Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable $C^*$-algebra such that $A''=M$. Let $p,q \in M_{\infty}(A)$ be ...
2
votes
2answers
305 views

Polynomials of low degree that clone polynomials of higher degree

Let $f(x_1,\dots,x_{16})=(x_1+x_2+x_3+x_4)(x_5+x_6+x_7+x_8)(x_9+x_{10}+x_{11}+x_{12})(x_{13}+x_{14}+x_{15}+x_{16})\in\Bbb R[x]$. Let $\mathcal{Z}$ be the zero set of $f$ in ...
8
votes
2answers
294 views

Dehn algorithm and normal forms in small cancellation groups

I found this statement in B. Cavallo, D. Kahrobaei's paper arXiv:1311.7117 Secret Sharing using Non-Commutative Groups and the Shortlex Order, page 7. "C′(1/6) continue to be an ideal platform for ...
6
votes
1answer
349 views

What is the universal property of quotienting a normaliser of the subgroup?

Let $G$ be a group, $H$ a subgroup and $X$ a $G$-set. By taking orbits $X/H = X \times_H 1$ or fixed points $X^H = \mathrm{Hom}_H(1,X)$ we obtain a set on which $H$ acts trivially, and we've destroyed ...
15
votes
2answers
369 views

ULU Decomposition of a matrix

Let $g \in GL_n(\mathbb{F}_q)$. Is it true that we can always write $g = u_1lu_2$, where $u_1$ and $u_2$ are upper-triangular and $l$ is lower-triangular? Note that I'm not requiring that the matrices ...
9
votes
1answer
375 views

Is a free group a product of f.g subgroups of infinite index?

Let $F$ be a free group, and let $H,K \leq F$ be finitely generated subgroups of infinite index in $F$. Is it possible that for the set of products we have $HK = F$ ?
3
votes
1answer
125 views

automorphism of prime order for group of Lie type in

Thanks for any help. Suppose $S$ is a simple group of Lie type of prime characteristic $p$. we know that every automorphism of $S$ is composite of inner, diagonal, field and graph automorphism of ...
5
votes
2answers
386 views

A generalisation of the theorem of Maschke

The theorem of Maschke tells us that every representation of a finite group is the direct sum of irreducible representation. More precisely: Let $G$ be a finite group, $K$ a field whose ...
3
votes
1answer
307 views

Reductive Lie Groups and Complexification

Let $G$ be a complex Lie group (not necessarily connected) with reductive Lie algebra $\frak{g}$. (We may assume that $G$ has finitely many connected components and is linear-algebraic.) Of course, ...
0
votes
0answers
56 views

“Reciprocal” of Schoenberg's theorem

Schoenberg's theorem states that for a (say, countable group) $G$ and any real valued conditionally negative type function $\psi$ on $G$, the function $e^{-t\psi}$ is positive definite, for any ...
5
votes
1answer
200 views

Is there a left orderable profinite group?

Is there a profinite group $G$ with a binary transitive relation $<$ such that for any different $x,y \in G$ either $x < y$ or $y < x$ and such that for any $x,y,z \in G$ we have that $x < ...
1
vote
0answers
87 views

Free abelian subgroups and distorsion

I realized that I know groups with distorted cyclic subgroups and groups all of whose free abelian subgroups are undistorted, but nothing between. Maybe it is a naive question, but: Does there ...
1
vote
1answer
82 views

A subgroup of outer automorphisms group of a free product

I would like to ask a question about automorphisms of free products of groups. More specifically, let $G = G_1 \ast ... \ G_n \ast F_r$ where $F_r$ is free group on r generators. We can define the ...
2
votes
0answers
87 views

Schur covering group [closed]

It is known that every finite group has a Schur covering group. I'm eager to know every finite group can be considered as a Schur covering group of a group. If it is not true in general, under what ...
0
votes
1answer
170 views

Coaction of a group

Suppose $G$ is a finite group which acts on a $C^*-$algebra which we denote by $A$. I was wondering if there is a naturally induced coaction on $A\otimes C(G)$, here $C(G)$ denotes functions on $G$. I ...
6
votes
2answers
324 views

A proposition on cyclic group

$G$ is a cyclic group iff $$ \forall H < G, \ \exists k, \ H = \{a^k : a \in G\}. $$ Is it right?
1
vote
1answer
48 views

Groups arising as direct limits of a stationary system of primitive matrices over the integers

I am interested in the kinds of groups of the form $\displaystyle\lim_{\longrightarrow}(\mathbf{Z}^k,M)$ where $M$ is a primitive (some power of $M$ has strictly positive components) $k\times k$ ...
26
votes
2answers
544 views

Why do sporadic simple groups have so few conjugacy classes?

In finite group theory, there's a general intuition that the further away a group is from abelian, the fewer conjugacy classes it will have. So it is to be expected that non-abelian finite simple ...
10
votes
1answer
162 views

Free subgroups in algebras of polynomial growth

What is known about free non-abelian subgroups in finitely generated associative algebras of polynomial growth (e.g., over finite fields, to avoid finite-dimensional free subgroups)? For example, are ...
5
votes
0answers
144 views

Approximating Lie groups by finite groups

How can one approximate compact Lie groups by finite groups? My wish is something like this: Let $G$ be a compact Lie group. There is a sequence of nested finite subgroups $G_n$ so that $G_n\to ...
0
votes
0answers
116 views

Comparison of two Chevalley basis

Let $G$ be a connected reductive group over an algebraically closed field and $T$ a maximal torus. Let $H$ be a pseudo-Levi subgroup, say the neutral component of a centralizer of a semisimple element ...
1
vote
1answer
123 views

Equation for non-invertible elements in Clifford algebras

Suppose we have a Clifford algebra $Cl(V,q)$, $V\simeq \mathbb{R}^n$ and $q$ non-degenerate bilinear form. Then every non-zero element of $V\subset Cl(V,q)$ invertible, but they are not the only ones ...
6
votes
1answer
234 views

Paper by I. N. Sanov, Solution of the Burnside problem for exponent 4

I have searched extensively online and for copies of printed journals containing the paper which details Sanov's solution to the Burnside Problem for exponent 4, which is widely cited in many papers ...
12
votes
1answer
681 views

Find finite groups $G\cong Aut(G)$

I become interested in this problem because $G\cong Aut(G)$ suggests a special symmetry in $G$ (This kind of group describes its own symmetry). From $G/Z(G)\cong Inn(G)$ we know complete group is the ...
7
votes
2answers
545 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 ...
1
vote
0answers
79 views

Subgroups of the union of conjugates

This question is an attempt to find a version of this question with a positive answer. In this question, we ask if one can find big subgroups in the union of conjugtes of small subgroups of finite ...
9
votes
4answers
564 views

Applications of n-dimensional crystallographic groups

I would like to know what are the applications of the theory of $n$-dimensional crystallographic groups (aka space groups) 1) in mathematics 2) outside of mathematics, besides the applications to ...
2
votes
1answer
165 views

Union of conjugates in p-groups

Fix a prime number $p$. Is there a sequence $\{a_k\}_{k \in \mathbb{N}}$ of real numbers with $$\lim_{k \to \infty} a_k = 0$$ such that for any finite $p$-group $G$, and any subgroup $H \leq G$ with ...
5
votes
1answer
310 views

Milnor-Wolf result on growth of solvable groups

The Milnor-Wolf theorem says that any solvable group has either polynomial or exponential growth. I wonder about the existence of alternative proofs of this fact. I have an impression that the ...
4
votes
1answer
123 views

orders of maximal abelian subgroups

What are the orders of maximal abelian subgroups of the simple groups $F_4(q)$ and $C_4(q)$, where $F_4(q)$ is an exceptional group and $C_4(q)$ is a symplectic group?
3
votes
2answers
184 views

Smallest non-trivial conjugacy classes in simple groups and classes of involutions

I am interested in finding the size of the smallest non-trivial conjugacy class of the simple groups $PSL(d,q)$ with $d>2$, $Sz(q)$ with $q>2$ and $R(q)$ with $q>3$. My first question is ...