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Questions about the branch of algebra that deals with groups.

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Real non-principal 2-blocks for finite groups of Lie type

Is it true that each finite group of Lie type has a non-principal real $2$-block? Here the principal block is the one containing the trivial character and a block is real if it contains the complex ...
8
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1answer
142 views

hyperbolic quotient of hyperbolic group

I have a memory of hearing about a result (or perhaps a conjecture), possibly due to Gromov, that, if $G$ is a hyperbolic group and $g \in G$ has infinite order, then the quotient group $G/\langle (g^...
0
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1answer
77 views

Isomorphism between two groups [closed]

Let $\mathbf{F}_q$ be finite field of order $q$, where $q$ is an odd-prime (or power of an odd prime). Is there an isomorphism between the following subgroups of unipotent $3\times 3$ matrices over $\...
-1
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0answers
65 views

What is the normal subgroup generated by nonsingular, positive definite matrices? [closed]

Within $GL(n,\mathbb C)$ let $H$ denote the normal subgroup generated by products of positive-definite, Hermitian matrices. What is this subgroup? And what is the quotient $GL(n,\mathbb C)/H$? Do ...
3
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1answer
63 views

Is the Singer cycle preserved by field automorphisms and graph automorphisms?

Let $T=\operatorname{PSL}_n(q)$ with $n$ a prime number. Then the $\mathscr{C}_3$ subgroup $M=\langle x\rangle{:}\langle\sigma\rangle$ of $T$ is isomorphic to $\mathbb{Z}_{\frac{q^n-1}{(q-1)(n,q-1)}}{:...
4
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0answers
56 views

Maximal $\mathbb{Q}$-split torus in center of Weil restriction of $\text{GL}_n$ over a number ring

I'm a topologist writing a paper where I have to do a bit of work with algebraic groups, and I've managed to confuse myself about something very basic. Let $K$ be an algebraic number field. Regard $\...
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0answers
57 views

Large gaps in the norm of a subgroup and its centraliser

Take an infinite finitely generated group $G$ with an infinite subgroup $N$ which has an infinite centraliser $Z = Z_G(N)$. Let $S$ be some [symmetric] generating set of $G$ and for $g \in G$, ...
6
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2answers
105 views

Minimal generation of simple groups and Ore's conjecture

The well known Ore's conjecture (now established) states that every element of a finite non-abelian simple group $G$ is a commutator of a pair of elements. Also we know that $G$ is $2$-generated. I ...
4
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0answers
153 views

Image of the mapping class group of surfaces into automorphism group?

Let $S_{g,p}^n$ be a compact oriented surface of genus $g$ with $p$ punctures and $n$ boundary components, and $\operatorname{Mod}(S)$ and $\operatorname{PMod}(S)$ be the mapping class group and the ...
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0answers
79 views

$G/F(G)$ is isomorphic to $X_1\times\cdots\times X_t$

I asked this question on math.stackexchange many hours ago, but haven’t got an answer. It was mentioned in a comment that the answer to my question is trivial, but I couldn’t see why. $G$ is a finite ...
1
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1answer
72 views

Is the solvable radical of a finite perfect group contained in the Schur multiplier of the quotient of the group modulo the solvable radical?

Let $G$ be a finite perfect group, and let $N$ be the solvable radical of $G$. If $G/N$ is a non-abelian simple group, then is it true that $N$ is contained in the Schur multiplier of $G/N$? If this ...
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0answers
61 views

QI-closure of $\mathrm{NA}\times\mathrm{NA}$

Suppose we know the following about a class of groups $\mathcal{G}$. If $G$, $H$ are f.g. r.p. nonamenable groups, then $G \times H \in \mathcal{G}$. If $G \in \mathcal{G}$, $G$ is f.p., and $G$ is ...
5
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1answer
164 views

Finite group ${\rm Sp}_4({\Bbb F}_3)$: involutions coming from a 4-dimensional complex representation

I am interested in the finite unitary reflection group $G= G_{32}$, the group No. 32 in Table VII on page 301 of the paper: Shephard, G. C.; Todd, J., A. Finite unitary reflection groups. Canad. J. ...
4
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1answer
114 views

Properties of the spectrum of the Koopman representation

Let $G$ be a discrete countable infinite group acting on a compact metric space $X$ via homeomorphisms preserving a probability measure $\mu$. A function $\lambda\colon G\to \mathbb C$ is an ...
6
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2answers
309 views

Representation of central extension

Let $G$ be a finite abelian group of rank $n$ and $H\rightarrow G$ a central extension with cyclic finite kernel. Is it true that we can find a faithful representation $H\rightarrow {\rm GL}_{k(n)}(\...
8
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0answers
126 views

Reference request: Name or use of this group of diffeomorphisms of the disc

Let $k \in \{0,\infty\}$, $G\subseteq \operatorname{Diff}^k(D^n)$ be the set of diffeomorphisms $\phi:D^n\to D^n$ of the closed $n$-disc $D^n$ (with its boundary) satisfying the following: $ \phi(S_r^...
3
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0answers
90 views

Are there perfect DTI-groups which are not simple?

Recall that a subgroup $H$ of a group $G$ is called a TI-subgroup if for every $g \in G$ it is $H \cap H^g \in \{1,H\}$. We say that a group $G$ is a DTI-group if the derived subgroups of all of its ...
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0answers
112 views

What can we get from an automorphism of order $2$

An automorphism of order $2$ is an automorphism fixes some elements and inverts the others. It’s well known that not all groups have automorphisms of order $2$, $C_2$, for example. But if a group ...
3
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144 views

Convergence of Fuchsian groups and existence of suitable homeomorphisms

Let $(\Gamma_n)_n$ ($\subset PSL(2,\mathbb{R})$) be a sequence of discrete groups, if we say that $(\Gamma_n)_n$ converges to a group $\Gamma$ this means that there exist isomorphisms $\tau_n:\Gamma\...
4
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109 views

Number of elements in $\mathrm{GL}(n,p)$ with maximal order

I learned reading this question that $\mathrm{GL}(n,p)$ elements have at most a multiplicative order of $p^n -1$. I would like to know how many matrices have an order of exactly $p^n -1$. Do they ...
3
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0answers
78 views

Residually amenable groups

I have two questions about residually amenable groups: Is every finitely presented amenable group residually elementary amenable? Is the free Burnside group on two generators residually amenable? ...
6
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1answer
398 views

Origin of the term 'index of a subgroup'

The index of a subgroup $H$ in a group $G$ is the number of distinct cosets of $H$ in $G$. Why did someone decide to call this an 'index'? What's the rationale for this?
8
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1answer
335 views

Conjecture by Ekedahl on Weyl groups and Abelian varieties

A conjecture was made on p.14 in "Cycle Classes of the E-O Stratification on the Moduli of Abelian Varieties" by Torsten Ekedahl (late, excellent contributor to MO) and Gerard Van Der Geer concerning ...
8
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0answers
117 views

Membership problem in general linear group

This is surely a very well known problem, but I could not find an answer on MO or on Google, so here I am. Given some finitely generated free subgroup $H$ of $\operatorname{GL}_n(\mathbb{Z}[t,t^{-1}])...
2
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1answer
85 views

Centralizer of a single element in the monoid of self-maps of a set

This is a follow-up to this question: For what sets $X$ do there exist a pair of functions from $X$ to $X$ with the identity being the only function that commutes with both? Let $X$ be a set, and $X^...
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0answers
60 views

Fitting subgroup of a solvable group is a direct product of some elementary abelian $p$-groups

I saw a remark saying If $G$ is solvable, then ${\rm Out}(F(G))$ is isomorphic to a direct product of ${\rm GL}(n_i,p_i)$ (except for certain value of $n_i$ and $p_i$). I know the key is to prove ...
2
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2answers
56 views

Upper density of subsets of an amenable group

Let $G$ be an amenable group (so locally compact Hausdorff) and also assume it is second countable if needed. My question is that what are the standard ways (across literature) of defining the upper ...
1
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1answer
219 views

injectivity of pushout?

We have the following pushout diagram: $$\begin{array}{ccc} \langle X, Y \rangle & \xrightarrow{\alpha} & \mathbb{Z}_a \ast \mathbb{Z}_b \ast \mathbb{Z}_c \ast \mathbb{Z}_d \\ \downarrow \...
10
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0answers
178 views

Roadmap to homotopical group theory

I have been lurking here for a long time just enjoying the scenery from my beginner's viewpoint. I have a math.SE account but I think this question is appropriate here based on the nature of the ...
2
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0answers
65 views

Finitely generated uniformly amenable groups

Keller in "Amenable groups and varieties of groups" introduces uniformly amenable groups as groups such that there is a function $a: ]0,1[ \times \mathbb{N} \to \mathbb{N}$ such that for any finite ...
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0answers
36 views

Intersection of subgroup of a free group with the lower central series

If I have a subgroup $S$ of a free group $\mathcal{F}_m$, what can I say about the behaviour of the descending sequence of subgroups $\left< S, \Gamma_c(\mathcal{F}_m) \right>$ (where $\Gamma_c(\...
2
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0answers
40 views

Can a polytope with vertex-transitive edge graph or face lattice be made vertex-transitive?

Let $P\subset\Bbb R^d$ be a convex, full-dimensional polytope (convex hull of finitely many points, affine hull is the whole space), $G_P$ its edge graph and $\mathcal F_P$ its face lattice. Any of ...
3
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1answer
74 views

Graph structure on $S_\omega$ induced by fixed points on compositions

Let $S_\omega$ denote the collection of bijections $f:\omega\to\omega$. We say that $f \in S_\omega$ has a fixed point if there is $x\in \omega$ with $f(x) = x$. It is a short exercise to show that ...
11
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0answers
308 views

Isomorphic free groups have bijective generating sets

Let $F(X)$ be the free group on a set $X$. Classically, we can prove the statement: $F(X) \cong F(Y)$ if and only if $|X|=|Y|$. The proofs (that I have seen) consist of turning the group ...
2
votes
1answer
200 views

Why $G/F(G)$ is isomorphic to a subgroup of ${\rm Out}(F(G))$?

I know two facts and I’ve managed to figure out how to prove one, but the other one is still a little confusing. Let $G$ be a finite solvable group and $F(G)$ is the Fitting subgroup of $G$. (...
4
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1answer
133 views

Finitely presented non-residually amenable groups without free subgroups

Does there exist a finitely presented group that does not contain a nonabelian free group and is not residually amenable?
5
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2answers
168 views

Seeking to understand meaning of “von Neumann spectrum” in a paper of Bader–Furman–Shaker

In attempting to understand the paper "Superrigidity, Weyl groups, and actions on the circle" of Uri Bader, Alex Furman and Ali Shaker (linked at Furman's page) I find that towards the end of the ...
6
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1answer
154 views

Why do we say the Fitting subgroup/generalized Fitting subgroup control the structure of a group?

I’m learning the Fitting subgroup these days. I’m interested in this topic and particularly in the role that it plays in the structure of groups. Many people on MSE mentioned that the Fitting subgroup/...
1
vote
1answer
66 views

Ergodic decomposition - how does restricting measure effect it? (Choquet Theory)

Suppose that $G$ is a discrete countable group and $\mu$ is an IRS (invariant random subgroup) of $G$: $\mu$ is conjugation invariant as a probability measure on the subgroups of $G$. Since all the $...
12
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1answer
192 views

Powers of the Euler class, torsion free subgroup of Homeo($S^1$)

For any subgroup $G$ of $\text{Homeo}(S^1)$, we have the Euler class $\chi$ in the group cohomology $H^2(G;\mathbb{Z})$. One can think of this class as the pullback of the generator of $H^2(\mathrm{B}\...
3
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1answer
114 views

action of symmetric group on the second exterior power

Let $e_i \wedge e_j \ (i < j)$ be a basis for the $\mathbb Z$-module $\wedge^2 \Gamma$, where $\Gamma = \mathbb Z^n$. Clearly $S_n$ acts on the module $\wedge^2 \Gamma$ via $$\pi(e_i \wedge e_j) ...
5
votes
1answer
105 views

Comparing $X+Y$ and $X-Y$ for independent random variables with values in an abelian locally compact group

Let $G$ be an abelian locally (separable?) compact group with Haar measure $\mu$. Inspired by the interesting proof of A sum of two binomial random variables : Let $X$ and $Y$ be $G$-valued ...
0
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0answers
49 views

Schur multiplier of 2-Sylow subgroups of symmetric group

Is the Schur multiplier of 2-Sylow subgroups of symmetric groups on $n\geq 4$ symbols known? I couldn’t find much except that multiplier of $S_n$ is contained in the multiplier of 2-Sylow subgroup.
0
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1answer
149 views

Conjugacy in the quaternion group

Let $G$ be a non-commutative group, and suppose we are given two elements $x, y \in G$ which are conjugate, i.e. we know there exists some $z \in G$ such that $zxz^{-1} = y$. Can we find $z$ given $x$ ...
3
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1answer
163 views

Groups whose non-linear irreducible characters are all non-faithful

I am interested in knowing if there is any literature that describes finite solvable groups whose non-linear complex irreducible characters are all non-faithful.
7
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2answers
163 views

Torsion group with finitely many elements of order 2 but infinitely many elements of order 4

Does there exists a group $G$ satisfying all the following conditions? $G$ is finitely generated, $G$ is of bounded torsion (has finite exponent), $G$ has finitely many elements of order $2$, $G$ has ...
2
votes
1answer
202 views

Recurrence relation for number of reduced words of longest element in $S_n$

Is there any recurrence relation known for the number of reduced words of the longest element in $S_n$ (not commutation classes)? Edit: Sorry for unaccepting the answer, but I realized that I really ...
20
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1answer
594 views

Is $A_5$ the only finite simple group with only 4 distinct sizes of orbits under the action of the automorphism group?

Given a finite group $G$, let $\eta(G)$ denote the number of distinct sizes of orbits on $G$ under the action of ${\rm Aut}(G)$. It happens that there are infinitely many non-abelian finite simple ...
3
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0answers
153 views

Hypothesis: An injection from polygons into $SO(2) \times S_n$

I have stumbled upon a possible representation of polygons by a concise description of their behaviour under rotation. I would like to know more about it and, obviously, if it is actually a bijection. ...
0
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0answers
70 views

What do Sylow 2-subgroups look like for Schur covering groups of finite simple groups?

What do Sylow 2-subgroups look like for Schur covering groups of finite simple groups? Are there any references in which we can find the stucture of Sylow 2-subgroups of Schur covering groups of ...

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