# Questions tagged [gr.group-theory]

Questions about the branch of algebra that deals with groups.

7,823
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### Show me that I have not simplified the proof of the Adian-Rabin theorem

I am not a mathematics researcher but I am concerned that this question, posed with slightly different wording on math.stackexchange, may be too esoteric for that forum since it concerns the details ...

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### short exact sequence of groups [closed]

We have a short exact sequence as
$$0 \rightarrow \mathbb{Z}_2\rightarrow G \rightarrow \mathbb{Z}_2\rightarrow 0,$$
can we conclude that the group $G$ is isomorphic to $\mathbb{Z}_2 + \mathbb{Z}_2$ ...

4
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1
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### Local triviality of torsors for relative reductive groups

Let $X \to S$ be a relative (smooth proper) curve, and $G \to X$ a reductive group scheme. The following two results are well-known:
(Drinfeld-Simpson) For arbitrary $S$, if $G$ is defined over $S$, ...

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1
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336
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### Does every transitive permutation group contain a permutation whose cycle lengths have a common divisor?

Let $H$ be a transitive subgroup of $\mathfrak{S}_n$, $n \geq 2$.
Using Jordan's lemma ($H$ is not a union of conjugate proper subgroups), we see that $H$ contains a permutation without fixed points.
...

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### What does this notation mean? [closed]

For context, $\phi_{f} $ and $\psi_{u} $ are two different automorphisms of the same group. I would like to know what the following notation, $\phi_{f}^{\psi_{u} } $ , is referring to? What does it ...

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### Can automorphism equivalence in a free group be detected in a nilpotent quotient?

If $G$ is a group and $g_1, g_2 \in G$ let us write $g_1 \sim g_2$ if there is an automorphism $\alpha \in \operatorname{Aut}(G)$ such that $g_1^\alpha = g_2$.
Let $F = F_2$ be the free group on two ...

12
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1
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475
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### Generators for the first cohomology of free groups

Let $F = \langle x_1, \dots, x_n \rangle$ be the free group on $n$ generators and $R = \mathbb Z$. The Fox derivatives $\frac{\partial}{\partial x_i} \colon F \to R[F]$ are the unique derivations ...

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### An identity for characters of the symmetric group

I am looking for a reference for the identity
$$\chi_\lambda(C)=\frac{\dim(V_\lambda)}{|C|}\sum_{p\in P_\lambda,\,q\in Q_\lambda,\,pq\in C}\operatorname{sgn}(q)$$
for the irreducible characters of the ...

12
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2
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578
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### The mysterious significance of local subgroups in finite group theory

EDIT 21/12: Even if there are no conclusive answers to these questions, I would very much like to know if anyone has noted and attempted to explain the mysterious significance of local subgroups: are ...

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### Generalisation of abelianisation using representation theory?

This question didn't receive an answer on MathSE, so I'm asking it here.
Let $G$ be a finite group and let $k$ be an algebraically closed field of characteristic zero. Every $1$-dimensional ...

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### What is the effect of tensoring with the sign representation on irreducible modules for a Type D Weyl group?

Given an integer $n \geq 4$, consider the Weyl groups $W(B_n)$ and $W(D_n)$ of types $B_n$ and $D_n$, respectively, and consider their representation theory over the field of complex numbers.
The Weyl ...

2
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1
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### When are these irreducible complex representations for the Type D Weyl group self-dual?

Given an integer $n \geq 4$, consider the Weyl groups $W(B_n)$ and $W(D_n)$ of types $B_n$ and $D_n$, respectively, and consider their representation theory over the field of complex numbers.
The Weyl ...

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### Example of three dimensional atoroidal Poincaré duality group with some pathology

I am looking for a 3-manifold which is closed, aspherical, orientable, and atoroidal. And additionally I want to see an example that does not admit a fixed-point-free action on a simplicial tree. As a ...

2
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1
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### Can a non-free Whitehead group embed as a discrete subgroup of a normed space?

Every countable discrete subgroup of a normed space is isomorphic to the direct sum of the group of integers. I wonder whether it is possible to push this beyond such direct-sum (free abelian) groups ...

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### What is known about the map $\text{Mod}_g^1 \rightarrow \text{Aut}(F_{2g})$?

Follow up question, edited in on 12/20 below:
Letting $\text{Mod}_g^1$ be the mapping class group of a surface with one boundary component (and basepoint on the boundary) and identify its fundamental ...

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### Large subgroups of Knuth's non-associative "group" on ${\cal P}(\mathbb{N})$

Donald Knuth introduced a fast, bit-wise approximation to integer addition by $$(a,b) \mapsto a \, ^{\land} \, b \, ^{\land} \, ((a \text{ & } b) \ll 1)$$
where $a,b$ are given in binary and $\,^{\...

3
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175
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### Commutator of a group element on a vector space

I am reading a paper in which the author has a group $G$ admitting a representation $\pi$ on a vector space $V$. Let $g \in G$ be a group element. The author refers to a so-called "commutator of $...

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140
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### Prove that every group $G$ with $p^n$ ($n\ge4$) elements and center with $p$ elements has an abelian subgroup of order $p^3$ [closed]

Prove that every group $G$ with $p^n$ ($n\ge4$) elements and center with $p$ elements has an abelian subgroup of order $p^3$
I'm new in this forum so I hope I haven't made any mistake.
I have to ...

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### Finitely generated G, such that x^3 = 1 for all x, is finite? [closed]

x^3 = e for any element x in finitely-generated group G. How to prove that G is finite?

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### Almost free group without the Specker group as a subgroup

An Abelian group is almost free whenever every countable subgroup is free Abelian. Famously, the Specker group $\mathbf Z^{\mathbf N}$ is almost free. What are examples of almost free groups that are ...

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1
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### Divergence functions in hyperbolic groups

Gromov hyperbolicity has many characterizations, one of them being the existence of a super-linear divergence function, see definition below.
We note that in $\mathbb{R}^2$ there is no divergence ...

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### Distances on spheres in Cayley graphs of non-amenable groups

Let $G$ be a non-amenable group (or perhaps more generally, a group with exponential growth). For any $\epsilon>0$, define the shell of radius r, $S_\epsilon(r)$, as the set of points that lie at a ...

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### Finitely generated torsion-free pro-$p$ subgroup of ${\rm GL}_{n}(\mathbb{F}_{p}[[T]])$ is solvable?

Let $\mathbb{F}_{p}$ be a finite field of order $p$, and $\mathbb{F}_{p}[[T]]$ be the ring of formal power series over $\mathbb{F}_{p}$. My question is the following:
Let $G$ be a closed pro-$p$ ...

7
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1
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### If two Lie algebras are isomorphic, under which conditions will their Lie groups also be isomorphic?

Let $G$ and $G'$ be compact connected Lie groups (which are not necessarily simply connected) with Lie algebra $\mathfrak{g}$ and $\mathfrak{h}$. Suppose that the two Lie algebras are isomorphic, ...

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### Explicit character tables of non-existent finite simple groups

In connection with the historical development of the classification of finite simple groups, I am interested in a particular aspect that seems to be less well-documented than the main narrative of ...

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1
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### Conjugacy of Cartan subgroups in $\mathrm{GL}(n)$

$\DeclareMathOperator\SL{SL}$I have probably a very basic question on the structure of semisimple Lie groups. Sorry if it is too elementary.
Let either $G=\SL(n,\mathbb{R})$ or $G=\SL(n,\mathbb{C})$. ...

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### Normalizer of the group of segment $C^\infty$ diffeomorphisms in the group of segment homeomorphisms

What is the normalizer of the group of $C^\infty$ diffeomorphisms on $[0, 1]$, with group law given by composition, in the group of all homeomorphisms of $[0, 1]$?
If the answer is known, is there ...

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### Normal subgroups of pure braid groups stable under strand bifurcation

$\DeclareMathOperator\PB{PB}\DeclareMathOperator\B{B}$Let $\PB_n$ be the $n$-strand pure braid group. For each $1\le k\le n$, let $\kappa_k^n \colon \PB_n \to \PB_{n+1}$ be the monomorphism that takes ...

6
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165
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### Finite groups with only one $p$-block

If $G$ is a finite group with a prime $p \big| |G|$, and $G$ has exactly one $p$-block, namely the principal block, can anything be said about the structure of $G$? I am aware that when $G$ has ...

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### Admissibility of Ulm's invariants

Let $G$ be a reduced abelian $p$-group. We set $G_0=G$. Let $\alpha$ be an ordinal. Inductively, if $\alpha=\beta+1$ is a successor ordinal, we define
$$G_{\alpha}=pG_{\beta}.$$
If $\alpha$ is a limit ...

2
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2
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170
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### Number of conjugacy classes of a semi-direct product of two finite groups

Let $G$ and $H$ be two finite groups. Let $r(G)$ be the order of the set of conjugacy classes of $G$. We know $$r(G\times H)=r(G)\times r(H).$$ My problem is: if there is a semi-direct product $G\...

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### Is $SO(N)$ a sphere? [closed]

Is $SO(N)$ as a metric space with a Killing form metric, isometric to a sphere $S^M$ (for any $M$) with standard metric on the sphere?

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### G-modules vs. $\Delta(NG)$-modules

Let X be a simplicial set. Its category of simplices, denoted by $\Delta(X)$, is the category whose objects are the pairs $(x,[n])$, with $x\in X_n$, and morphisms $\bar{c}:(y,[m])\to (x,[n])$, where $...

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### Recognizing free groups

While I’m aware (and as it was pointed out in the comments) it is generally undecidable whether a given presentation represents a free group, I’m interested in criteria that nevertheless ensure that ...

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### A conceptual proof that bounded index subgroups of a bounded torsion abelian group contain bounded index complemented subgroups

Call an abelian group $G = (G,+)$ $m$-torsion for some natural number $m$ if one has $m \cdot x = 0$ for all $x \in G$. A subgroup $H$ of $G$ is said to be complemented if one can write $G = H \oplus ...

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### Orthogonal representation of free products of two groups

Suppose $A$ and $B$ are two countable, discrete, amenable groups. One definition of amenability tells us that there is a sequence of finitely supported, positive definite functions that converges to 1 ...

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### Research directions related to the Hilbert-Smith conjecture

The Hilbert-Smith Conjecture (HSC) is a famous open problem in geometric group theory stating "for every prime $p$ there are no faithful continuous action of the $p$-adic group of integers $A_p$ ...

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### Do balls in expander graphs have small expansion?

Consider a $d$-regular infinite transitive expander graph $G$, and let $B_r$ be a ball of radius $r$ in $G$. Can one place any upper bounds on the expansion of $B_r$?
My intuition is that $B_r$ will ...

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### Groups (not necessarily finite) with a given number of maximal subgroups

It is somewhat easy to see that a group $G$ with exactly one maximal subgroup $M$ must be cyclic: any element in $G\setminus M$ generates $G$.
EDIT: @YCor pointed out in the comments that this ...

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### Salvetti complex of dihedral Artin group

The Salvetti complex of a RAAG is well-known and it is fairly simple, since each complete graph gives rise to a tori. The case of Artin groups is wilder, since we do not have tori anymore. The ...

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### Is this set, defined in terms of an irreducible representation, closed under inverses?

$\DeclareMathOperator\GL{GL}$Let $ H $ be an irreducible finite subgroup of $ \GL(n,\mathbb{C}) $. Define $ N^r(H) $ inductively by
$$
N^{r+1}(H)=\{ g \in \GL(n,\mathbb{C}): g H g^{-1} \subset N^r(H) \...

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### Units of the group algebra of a free group

Let $K$ be a field of characteristic zero and $F_n$ be a free group of rank $n$. What is known about the group of units $K[F_n]^\times$?
In the case of $n=1$, there are only trivial units: $K[F_1]^\...

4
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4
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### Groups acting non-properly cocompactly on hyperbolic spaces

A group $G$ is hyperbolic if it admits a geometric (the action is proper and co-bounded) action on a geodesic hyperbolic metric space. Also, the definition can be given as follows, a group $G$ ...

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### Hereditarily just-infinite pro-$2$ groups

An infinite profinite group $G$ is called just-infinite if all non-trivial closed normal subgroups of $G$ have finite index. A profinite group is called hereditarily just-infinite if every open ...

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### How to conclude the quasi-projective case of the derived McKay correspondence from the projective case?

I am currently trying to understand the paper "Mukai implies McKay" from Bridgeland, King and Reid (cf. here). Let me sum up the setting we find ourselves in:
Let $M$ be a smooth quasi-...

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2
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### Non-simple groups $G$ with only non-trivial quotient isomorphic to $G$

If $G$ is a group such that every non-trivial subgroup is isomorphic to $G$ itself, then $G= \mathbb{Z}$ is the only infinite group with that property (up to isomorphism). Amongst the finite groups we ...

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### Dimension inequality for primary groups

Let $p$ be a prime number and $G$ an abelian group.
The group $G$ is said to be $\textbf{primary}$ if every element of $G$ has order power of $p$.
For every natural number $n$, we define
$$\ker(p^n)=\{...

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### Does every faithful action on a scheme act freely on a dense open subset?

Disclaimer: I have asked this question on math exchange a week ago (here), but sadly to no avail. So I decided to escalate my question:
Let $G$ be a finite group acting faithfully on a smooth quasi-...

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### Bochner theorem for (non-abelian) discrete groups

I am interested in Pontryagin duality-like theories for discrete groups, more particularly, whether an analogue to Bochner's theorem for abelian groups exists in the discrete non-finite and non-...

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### Is $\text{Sym}(\omega)/\text{(fin)}$ embeddable in $\text{Sym}(\omega)$? [duplicate]

Let $\omega$ denote the set of natural numbers, let $\text{Sym}(\omega)$ be the collection of bijections $\psi:\omega\to\omega$, and let $\text{(fin)}$ be the set of members of $\text{Sym}(\omega)$ ...