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

**6**

votes

**1**answer

137 views

### Is there a highly transitive action of a finitely generated torsion simple group?

Is there a highly transitive action of a finitely generated torsion simple group $G$ on $\mathbb{Z}$ ?
Highly transitive means $k$-transitive for each $k \in \mathbb{N}$, that is: for every two ...

**1**

vote

**1**answer

125 views

### Does the hyperoctahedral group have only 3 maximal normal subgroups?

An hyperoctahedral group $G$ is the wreath product of $S_2$ and $S_n$, where $S_{n}$ is the symmetric group on $n$ letters, or in other words the semi-direct product $G=S_2^n\rtimes S_n$, w.r.t. the ...

**5**

votes

**1**answer

162 views

### What is the Schur multiplier of the affine linear group AGL(n,q)?

What is the Schur multiplier of the $n$-dimensional affine linear group $\mathrm{AGL}(n,q)$ over the Galois field with $q$ elements?
I am particularly interested in the simple case $n=1$. Computation ...

**4**

votes

**1**answer

243 views

### Generating infinite index subgroups of a free group

Let $F$ be nonabelian finitely generated free group, let $H \leq F$ be a finitely generated subgroup of infinite index and let $x,y \in F \setminus H$. Must there be some $a \in F$ such that $[F : ...

**7**

votes

**3**answers

215 views

### Polarizations generate the ring of invariants?

The symmetric group $S_n$ acts on $\mathbb R^n$ by permuting the coordinates and the ring of polynomial invariants is generated by the elementary symmetric polynomials. If we restrict the action to ...

**4**

votes

**1**answer

145 views

### A small rank linear combination of a small number of elements of a group

This is a version of
this question of Klim Efremenko.
Let $r>2$ be a natural number, say $r=3$ or $r=10$. Let $G$ be a finite group and $\rho$
be an irreducible complex representation of $G$. We ...

**3**

votes

**1**answer

131 views

### Co-rank of a group with $a^2b^2c^2=1$ (fundamental group of non-orientable surface)

What is the co-rank of a group $$G=\langle a_1,a_2,\dots,a_h\mid a_1^2a_2^2\dots a_h^2=1\rangle,$$ that is, finitely generated group with $h$ generators and one relation?
By co-rank, I mean the ...

**4**

votes

**1**answer

207 views

### Characterizing residually amenable groups

Let $G$ be a finitely generated group. The amenability of $G$ is equivalent to the existence of a certain "weak measure" on $G$. Is there such a characterization for residually amenable groups as ...

**3**

votes

**1**answer

214 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 ...

**5**

votes

**0**answers

81 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 ...

**5**

votes

**0**answers

164 views

### Division algebras over extension fields / reducibility of $G$-modules

Reformulation of the question (see below for the original question): Let $K$ be an algebraic number field and $D$ a finite-dimensional $K$-division algebra. Is there a description of the field ...

**2**

votes

**1**answer

84 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 $H$ ...

**3**

votes

**2**answers

158 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 ...

**1**

vote

**1**answer

180 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

**0**answers

141 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 ...

**3**

votes

**0**answers

147 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 ...

**7**

votes

**1**answer

248 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

**0**answers

80 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 ...

**0**

votes

**1**answer

63 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 ...

**6**

votes

**1**answer

252 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

**1**answer

128 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

**2**answers

284 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 ...

**0**

votes

**1**answer

138 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 ...

**9**

votes

**1**answer

384 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

**1**answer

132 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 ...

**15**

votes

**2**answers

392 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 ...

**0**

votes

**0**answers

60 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 ...

**1**

vote

**0**answers

92 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 ...

**2**

votes

**0**answers

90 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 ...

**1**

vote

**1**answer

87 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 ...

**6**

votes

**2**answers

332 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?

**2**

votes

**2**answers

317 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 ...

**6**

votes

**1**answer

362 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 ...

**0**

votes

**1**answer

177 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 ...

**1**

vote

**1**answer

61 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) $k\times k$ ...

**5**

votes

**1**answer

203 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 < ...

**26**

votes

**2**answers

579 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

**1**answer

165 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

**0**answers

156 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 ...

**1**

vote

**1**answer

127 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 ...

**0**

votes

**0**answers

127 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

**0**answers

80 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 ...

**2**

votes

**1**answer

169 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 ...

**4**

votes

**1**answer

124 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

**1**answer

108 views

### Lower Central Series of Pure Braid Groups?

What is the lower central series $\Gamma_k(P_n)$, where $P_n$ is the pure braid group with $n$ strands? We know that $P_n$ is generated by elements $A_{i,j}$; do we know the generators of ...

**6**

votes

**2**answers

385 views

### What are the outer automorphisms of a Coxeter group?

I want to know the outer automorphisms of the Weyl group of $\mathrm{E}_8$, if any.
But I might as well ask the question more generally. Suppose we have a Coxeter diagram. This gives a Coxeter ...

**6**

votes

**1**answer

244 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 ...

**3**

votes

**2**answers

186 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 ...

**6**

votes

**1**answer

155 views

### Is the free abstract group residually of rank d > 2?

Let $d \geq 2$ be an integer, and let $\mathcal{F}_d$ be the family of finite groups such that $G \in \mathcal{F}_d$ if and only if every subgroup of $G$ can be generated by at most $d$ elements.
Is ...

**4**

votes

**0**answers

141 views

### Is there a nontrivial profinite word which is trivial in any group with at most d generators?

Let $F$ be a free profinite group of rank $\aleph_0$, and let $d \in \mathbb{N}$.
Let $N_d \lhd_c F$ be the intersection of all open normal subgroups $L \lhd_o F$ for which $F/L$ can be generated by ...