# Questions tagged [gr.group-theory]

Questions about the branch of algebra that deals with groups.

7,986
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### A question about coprime automorphisms of profinite groups

Let $p$ a prime. A finite group is a $p'$-group if its order is prime to $p$. Let $A$ be a finite $p'$-group of automorphisms of a finite $p$-group $G$. Suppose that $A$ is a non-cyclic abelian group. ...

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### Definition of free profinite product of infinitely many groups

If we have profinite groups $G_1,...,G_n$ we can define its free profinite product in the natural way. But this natural definition (similar to the abstract case but in the category of profinite groups)...

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### Infinite closed subgroup of ${\rm SL}_{n}(\mathbb{F}_{p}[[T]])$ with full residual image

Let $\mathbb{F}_{p}$ be a finite field of order $p$, $\mathbb{Z}_p$ be the ring of $p$-adic integers and $\mathbb{F}_{p}[[T]]$ be the ring of formal power series over $\mathbb{F}_{p}$. For $p\geq 5$, ...

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### Model-theoretic construction of Gromov boundaries on groups

For context, I'm only a second year undergraduate mathematician, so I won't know much.
For third year, I'm hoping to do a research project. I met up with a professor who might be my supervisor today, ...

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### $M^3$ admits $Sol$ geometry if and only if $\pi_1M$ is virtually solvable but not virtually nilpotent?

Let $M$ be a closed, orientable, irreducible 3-manifold and having an infinite fundamental group. Is it true that $M$ admits $Sol$ geometry if and only if $\pi_1M$ is virtually solvable but not ...

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85
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### Can we find background noise for every Følner sequence in a countable amenable group?

Let $G$ be a countable amenable group. We consider sequences $(z_g)_{g\in G}$ of complex numbers with $|z_g|=1$ for all $g\in G$.
I will say $(z_g)_{g\in G}$ is background noise for a (left-)Følner ...

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1
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### The stabilizer of a point in the connected Lorentz group

$\DeclareMathOperator\SO{SO}$For $n\geq 2$, the $n$-dimensional connected Lorentz group is defined as $$\SO(n-1,1)^{\uparrow}=\{f\in M_n (\mathbb{R}):f^T\eta f=\eta, \det (f)=1,f^0_0>0\}$$ where $$...

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652
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### Does some knot group fail to surject onto any alternating group?

Are there known examples of knot groups that do not surject onto any alternating group? I would be a little surprised if the answer is negative.
Update: To reflect the interesting part of the question,...

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523
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### Generators of a group and normal subgroups

Can we say anything about a minimal generating set of a finite group based on its normal subgroups? For example, can we bound their order, or say whether they come from the same conjugacy class?
An ...

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147
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### Hyperoctahedral group, preliminaries [closed]

I am looking for information on the hyperoctahedral group
From what I understand, the hyperoctahedral group is 'the generalized symmetric group' in the case where $m=2$. That is, the hyperoctahedral ...

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### An unpublished calculation of Gauss and the icosahedral group

According to p. 68 of Paul Stackel's essay "Gauss as geometer" (which deals with "complex quantities with more than two units") , Gauss calculated the coordinates of the vertices ...

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123
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### Density of numbers where a large prime factor satisfies a congruence

I am looking for an upper bound on the number of integers $n<x$ such that $n$ has a prime factor $p>\log(x)^{(1+\delta)}$ such that $p \equiv a \mod b$. Where $a,b$ are fixed and coprime and $0&...

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169
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### On Suzuki 2-groups being 2-Sylow subgroups

$\DeclareMathOperator\Sz{Sz}\DeclareMathOperator\SU{SU}$Recall the definition of the finite Suzuki 2-groups: These are finite non-abelian 2-groups that contain more than one involution such that a ...

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141
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### Centers and conjugacy classes of groups relative to a pair of group homomorphisms

$\newcommand{\defeq}{\mathbin{\overset{\mathrm{def}}{=}}}$Given a group $G$, its center $\mathrm{Z}(G)$ and set of conjugacy classes $\mathrm{Cl}(G)$ are defined by
\begin{align*}
\mathrm{Z}(G) &\...

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### What is the name of the largest subobject where a map is equivariant?

Suppose we have two objects $X,Y$ with a $G$-operation and a non-equivariant map between them. In this situation, we can look at the largest subobject $X'$ of $X$ on which $f$ is $G$-equivariant.
Is ...

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### Tangent space of Lie group SO(n) [migrated]

I have a potentially simple question here, about the tangent space of the Lie group SO(n), the group of orthogonal $n\times n$ real matrices (I'm sure this can be asked more generally). For any $R\in \...

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### Crossed homomorphism as morphism in the ambient category

Suppose we are given a crossed-homomorphism $\phi:G\to A$ (and an action $\alpha$ of $G$ on $A$)
$\phi(ab)=\phi(a)+\alpha(a)(\phi(b))$. Now, unless the action is trivial, this is not a homomorphism ...

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2
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### Maximal subgroups of finite abelian $2$-groups

Suppose $G$ is a finite abelian $2$-group, and $S$ is a subset of $G$, $\langle S\rangle=G$,$S^{-1}=S$,$e\notin S$. How to determine whether there exists a maximal subgroup $M$ of $G$, such that $S$ ...

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440
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### Are Artin-Tits groups ordered groups?

We consider Artin-Tits groups of two generators $(I_2(n))$. Are these groups ordered groups?

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### H^2 of symmetric group

I'm a number theorist in need of some group cohomology lemmas, and I'm rather bewildered by the level of generality used in the literature. Specifically, the result I need is as follows: the ...

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### What are the possible symmetry groups of n-point constructions in the projective plane?

Let $k$ be an infinite field, perhaps take $k = \mathbb{C}$ if it simplifies matters.
I will be asking a question about $\mathbb{P}^2$ for definiteness and to simplify definitions/notations, but feel ...

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### On dual notions of morphisms of algebraic structures obtained by replacing equaliser with coequalisers

This question is based on this discussion from the Category Theory Zulip. See also the earlier question Natural cotransformations and "dual" co/limits.
Let $G$ and $H$ be groups. We define ...

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### Necessary and sufficient conditions for the Cayley graph to be bipartite

Let $G$ be a finite group with identity $1$. If $S$ be an inverse closed generated subset of $G$, then $S$ is called a Cayley subset of $G$.The Cayley graph $\Gamma=\operatorname{Cay}(G, S)$ is a ...

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170
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### The existence of solutions to linear systems of equations over the integer ring $\mathbb{Z}$

There are already detailed results on the solutions of linear equations over fields, but I'd like to inquire about any good conclusions regarding the solutions of linear equations over the integer ...

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### Relativisation of Higman's embedding theorem

Higman's embedding theorem says that any finitely generated recursively presentable group is embeddable in a finitely presentable group. The converse is also true (a finitely generated subgroup of a ...

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### Integral homology of groups with finite exponent

I'm interested in the homology of infinite groups and especially in low-dimension integral homology.
If $G$ is a locally finite group of finite exponent, one has that also $H_*(G;\mathbb{Z})$ has ...

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1
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234
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### Commutator group and conjugacy classes

Let $G$ be a finite solvable group which is not nilpotent, and let $H=[G,G]$ be the commutator subgroup of $G$. Does the following hold for $G$ and $H$?
"There exists $g \in G \setminus H$ and $h ...

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### Point stabilizers of the Floyd boundary of a group

Let $G$ be a finitely generated group. Consider the Floyd boundary as defined in https://www.unige.ch/math/folks/karlsson/free.pdf by A. Karlsson. For a Floyd function f, we denote the Floyd boundary ...

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### A partial order on conjugacy classes

My esteemed colleague Nathan Bowler has drawn my attention to the
following partial order. Let $G$ be a group of permutations of an
infinite set $X$. If $\sigma$ and $\tau$ are elements of $G$ we say
...

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2
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### Groups whose derived length is logarithmic in the order?

Is there a class of solvable groups $G$ having a derived length $O(\log\lvert G\rvert)$?
See Wikipedia for the definition of Big-Oh ($O$) and the definition of derived series of a group.
Any help ...

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### *-Representation of a discrete group as a sum of cyclic representations

A cyclic representation on a Hilbert space is one where there exists a vector such that, under the action of the representation, this vector spans the entire Hilbert space.
It is known that every *-...

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### Alternating bihomomorphism is a skew 2-cocycle

It seems to be a well-known fact that every alternating bihomomorphism $G\times G\to\mathbb{C}^\times$ for a finite abelian group $G$ is the skew of some 2-cocycle (see for instance Symmetric analogue ...

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### Conjecture about semigroups

Let $G$ be a finite semigroup with order $n$ odd. Let $S_i \in G, i=1,\ldots,\binom{n}{(n+1)/2}$ be all the subsets of $G$ of size $(n+1)/2$.
Let $E(S_i)$ be the set obtained "expanding" $...

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### A stronger(?) notion than uniform contractibility

Let's call a metric space $ X $ strongly contractible if there exists a
function $ \rho : \mathbb{R}_+ \to \mathbb{R}_+ $ such that for every ball $
B(x;r) $ around a point $ x \in X $ we have:
$ B(x;...

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### Largest primitive subgroup of $\mathrm{GL}_8(\mathbb{C})$ of order $2^a 3^b 5^c$

The paper "Bounds for finite primitive complex linear groups" by M. Collins computes the largest possible value of $[G:Z(G)]$ for $G$ a primitive subgroup of $\mathrm{GL}_N(\mathbb{C})$, for ...

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### When $G$ is amenable is true that for a set $A \in p$ which is left piecewise syndetic then $\{ x: Ax^{-1} \in p \}$ is both sided syndetic?

Let be $G$ be a discrete group. I recall the definition of syndetic sets, thick sets and piecewise syndetic sets.
Definitions:
A set $A$ is left syndetic if there exist a finite $H \subset G$ such ...

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377
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### Number of finite groups: is $\operatorname{gnu}(4n) \geq 2 \operatorname{gnu}(n)$?

Let $\operatorname{gnu}(n)$ be the number of finite groups of order $n$.
Question: Is it true that $\operatorname{gnu}(4n) \geq 2 \operatorname{gnu}(n)$ for all $n \geq 1$?
Surely this must be true, ...

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### Normalizer of Levi subgroup

Let $G$ be a reductive group (we can work on an algebraically closed field if needed) and let $L$ be a parabolic subgroup, i.e. the centralizer of a certain torus $T \subseteq G$.
Associated with this ...

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### Involution of $\text{GL}_{m+n}(\mathbb{C})$ fixing Levi and exchanging parabolic subgroups

Is there any involution of $\text{GL}_{m+n}$ which is the identity on $\text{GL}_m\times\text{GL}_n\subset\text{GL}_{m+n}$ and that exchanges the positive and negative associated parabolic subgroups $...

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### Non-monotileable amenable groups

This is crossposted from MSE.
We say a subset $A$ of a group $G$ is a monotile for $G$ if $G$ is a disjoint union of right translates of $A$.
In his article Monotileable Amenable Groups, B. Weiss ...

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### Is Malcev completion an embedding?

The Malcev completion of an abstract group $G$ over a field $k$ of characteristic zero is defined by
$$\hat G = \mathbb{G}(\widehat{k[G]}) ,$$
the group-like part of the completed (by the augmentation ...

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### How might someone with a background in group theory start research into topos theory?

The Question:
How might an early career mathematician with a background of research in group theory start research into topos theory?
I want links between the two areas, not career advice, though it ...

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147
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### Up to what order have finite groups been classified? [duplicate]

All finite simple groups have been classified, and the classification of finite groups is thought to be wild. So, up to what order have finite groups been classified? Wikipedia tells us that it is ...

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### Finite group extensions of lattices

I'm currently reading the proof of Geroch's conjecture in Lawson-Michelsohn's Spin Geometry book and in the proof of Proposition IV.5.8 that every Ricci-flat enlargeable manifold is flat the following ...

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### Complex reflection groups: reference request

Suppose that $V$ is a finite-dimensional complex vector space, that $m\ge 2$ is an integer and that $G\subset \operatorname{GL}(V)$ is a finite subgroup such that $V$ is an irreducible ${\mathbb{C}}[G]...

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### On intersection of finite index fully invariant subgroup

Let $G$ be a group. A subgroup $H$ of $G$ is said to be fully invariant if for every endomorphism $\phi $ of $G$, we have $\phi(H) \subseteq H$. For a finitely generated residually finite group $G$, ...

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### Amplification argument for hyperlinear groups

Let us define a group $G$ to be a hyperlinear group if it satisfies the conclusion of Theorem 3.6. in these notes by Vladimir Pestov. It is well-known that one can use the so-called amplification ...

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### Separability in Coxeter groups

I am looking for a reference for the following statement:
Theorem. Let $\Gamma$ be a finite labelled graph and $C(\Gamma)$ the corresponding Coxeter group. For every $\Xi \subset V(\Gamma)$, the ...

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### Connection between certain finite groups and Frobenius algebras

This question is maybe not so precise yet, but contains some ideas that maybe just search for the right definition.
Let $G=F/U$ be a finite group with presentation $F/U$ (here the presentation really ...

9
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3
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846
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### Finding a nilpotent, infinite, f.g., virtually abelian, irreducible integer matrix group

I was wondering if anyone here knows of an example of a group $M \leq \mathrm{GL}_n(\mathbb{Z})$ which is
nilpotent,
infinite,
finitely generated,
virtually abelian,
irreducible (over $\mathbb{Z}$ or ...