All Questions
Tagged with group-theory or gr.group-theory
8,181 questions
2
votes
1
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219
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Existence of a special ordering of the elements of a finite group (II)
Let $G$ be a finite non-abelian group of order $n$. Given $g\in G$ we denote its order by $\mathrm{ord}(g)$.
Consider the group algebra $\mathbb{F}[G]$ for some field $\mathbb{F}$.
Given an ordering $...
- 14.6k
8
votes
2
answers
367
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Existence of a special ordering of the elements of a finite group
Let $G$ be a finite non-abelian group of order $n$. Given $g\in G$ we denote its order by $\mathrm{ord}(g)$.
Consider the group algebra $\mathbb{F}[G]$ for some field $\mathbb{F}$.
Given an ordering $...
- 14.6k
1
vote
0
answers
93
views
Exploring Plancherel measure decay rates linked to a specific $AD(\Gamma)$ range
In this paper on the amenability constant of Fourier algebras Theorem 1.5 presents a formula connecting $AD(\Gamma)$, the anti-diagonal constant of a countable virtually abelian group $\Gamma$, to ...
4
votes
2
answers
313
views
Structure of Sylow $p$-subgroup of $G$ with given property
Let $G$ be a finite group and $A\leq \operatorname{Aut}(G)$. Assume that $P$ is a Sylow $p$-subgroup of $G$ and $z\in Z(P)$ of order $p$ such that each subgroup of order $p$ of $P$ is $A$-conjugate ...
- 173
2
votes
0
answers
157
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Cohomologically trivial modules over finite $p$-groups
Let $A$ be a finitely generated $\mathbb{Z}_pG$-module, where $G$ is a finite $p$-group and $\mathbb{Z}_p$ is the ring of $p$-adic integers; assume moreover that $A$ is cohomologically trivial, that ...
4
votes
1
answer
318
views
Can $\text{Aut}(G)$ be extended to contain $G$?
Let $G$ be a group (finite, say) with center $Z$. The automorphism group $\text{Aut}(G)$ sits in a short exact sequence
$$ 1 \to G/Z \to \text{Aut}(G) \to \text{Out}(G) \to 1. $$
So when $Z\neq 1$, as ...
- 815
1
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0
answers
86
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Is the Galois closure of a $p$-adic Lie group extension also a $p$-adic Lie group extension?
Let $p$ be a prime. Let $K$ be a number field and $L/K$ be an infinite extension which is not necessarily Galois. Suppose that the automorphism group $\text{Aut}(L/K)$ of $L/K$ is $p$-adic analytic, i....
- 667
6
votes
1
answer
79
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Are finitely discriminable groups the compact elements of the poset of marked groups?
A finite discriminating family in a group $G$ is a finite set of non-identity elements such that every non-trivial normal subgroup of $G$ contains one of these elements.
A $k$-marked group is a group ...
- 543
1
vote
0
answers
66
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mod p cohomology of a p-group P vs. the one of P/Z(G)
Let $P$ be a p-group. Denote by $Z(G)$ its center and $H^*(P,\mathbb{F}_p)$ its mod $p$ cohomology ring.
My general (vague) question is what can be said about the mod $p$ cohomology of the central ...
5
votes
2
answers
443
views
Series of discrete groups with a Lie group limit
The groups ${\mathbb Z}_N$ may be viewed as a series, $N=1,2,3,\ldots$, which in the limit $N\to\infty$ approaches $U(1)$. I realize this is a bit hand waving but I'm pretty sure it can be made ...
- 1,150
1
vote
0
answers
155
views
Schreier's theorem for non-abelian cohomology
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Ext{Ext}$Let $A$ by $G$ be two non-abelian group. A factor pair of $G$ with coefficients in $A$ is a pair $(\alpha,\varepsilon)$, where $\alpha:G\...
- 393
4
votes
1
answer
259
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Howson's property for amalgams of free groups
Recall that a group has the Howson property (also known as the finitely generated intersection property) if the intersection of any two finitely generated subgroups is again finitely generated.
I am ...
- 51
4
votes
1
answer
266
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Are (group theoretic) Markov properties on groups with decidable word problems, decidable?
(Link to SE duplicate: https://math.stackexchange.com/questions/4959071/are-group-theoretic-markov-properties-on-groups-with-decidable-word-problems)
The Adian-Rabin theorem says that if a property of ...
- 191
5
votes
1
answer
257
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Finite simple $\pi$-groups
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$Let $\pi$ be a finite set of primes. A finite group $G$ is a $\pi$-group if all primes dividing $|G|$ lie in $\pi$. Is it true that there ...
- 56.9k
2
votes
1
answer
232
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Reference for surjectivity of the canonical map $R^{G_1} \otimes R^{G_2} \to R^{G_1 \cap G_2}$
Let $R$ be a commutative ring, $G$ a finite group with an action over $R$. Let $G_1, G_2 \subset G$ be two subgroups. Then the canonical map $R^{G_1} \otimes R^{G_2} \to R^{G_1 \cap G_2}$ is ...
- 23
2
votes
0
answers
82
views
Product of a standard parabolic subgroup with the opposite one
Let $G$ be a semisimple algebraic group defined over an algebraic closed field. Fix a Borel subgroup $B$ and a maximal torus $T\subset B$. Let $\Delta$ be the set of simple roots. For a subset $\...
- 653
13
votes
1
answer
420
views
Embedding rank of finite groups and quotients
Let $G$ be a finite group, and $n$ a positive integer. It is not hard to check that the following are equivalent:
For every $g\in G\setminus\{1\}$ there is a subgroup $H\leq G$ with $|G/H|\leq n$ ...
- 56.9k
3
votes
1
answer
107
views
Polynomial isoperimetric inequalities for finitely presented subdirect products of limit groups
Does every finitely presented subdirect product of limit groups admit a polynomial isoperimetric inequality? That is, does there exist a constant $C > 0$ and an integer $d \geq 1$ such that for ...
- 39
7
votes
1
answer
226
views
When is the action of a mapping class group on the set of punctures realized by a finite subgroup of mapping classes?
I have a curious question about a natural sequence, which I haven't seen answered in the literature.
Let $\Sigma$ be an oriented surface of genus $g$ without boundary with a set $\mathcal P_n$ of $n$ ...
- 318
3
votes
0
answers
161
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Generalized dimension property for rings
My question is very basic, I am looking for a characterization (and name) of rings $R$ satisfying the following property $\star$.
For any $V, W$ two finitely generated $R$-modules such that $V\oplus W\...
- 223
3
votes
0
answers
87
views
Stem extensions and quotients of Schur covers
Suppose that $G$ is a finite group, and that $\Gamma$ is a central extension of $G$ by $A$, that is
$$ 1 \rightarrow A \rightarrow \Gamma \rightarrow G \rightarrow 1$$
with the image of $A$ contained ...
- 1,300
7
votes
1
answer
328
views
A projectivity property in the category of groups
Let $F_r$ be the free group on $r$ generators, let $G$ and $H$ be finite groups, and let $F_r\xrightarrow{\alpha}H\xleftarrow{\beta}G$ be surjective homomorphisms. It is then easy to see that we can ...
- 56.9k
2
votes
1
answer
188
views
Higher Bockstein maps in group cohomology
Let $p$ be an odd prime and $n>1$. I am trying to understand why the cohomology ring $H^{\ast}(\mathbb{Z}/p^n;\mathbb{F_p})$ is given by
$$\mathbb{F}_p[y]\otimes\Lambda(x),$$
with $|x|=1,|y|=2$ and ...
- 245
12
votes
1
answer
539
views
Outer semidirect product
Suppose we have a group $G$ and two homs $L, R: G \rightarrow G$ such that $L^2 = L$, $R^2 = R$, and $LR = RL$. The first two conditions tell us that $L$ and $R$ are retractions followed by their ...
- 591
7
votes
2
answers
1k
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Groups killed by centralizing one element
What groups $G$ contains an element $g$ such that $G/(g\text{ is made central})=1$ (or equivalently $[g,G]=G$, where $[g,G]:=\langle[g,h]\colon h\in G\rangle$)?
A necessary condition is that $G$ is a ...
- 557
1
vote
0
answers
125
views
Generators of a Coxeter group
Let $(W,S)$ be a Coxeter system and assume that $|S|$ is finite. Certainly, $W$ is generated by $|S|$ simple reflections. My question is: Can $W$ be generated by fewer reflections? (Including non-...
- 71
1
vote
0
answers
84
views
Can a limit of degenerate two-cocycles be non-degenerate?
Let $G$ be a discrete abelian group and $\omega\colon G\times G\to\mathbb{T}$ a two-cocycle on $G$. We say that $\omega$ is non-degenerate if for every $e\neq g\in G$ there exists $h\in G$ such that $\...
- 29
7
votes
1
answer
259
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A name for the Weyl group of $\frak{so_{2n}}$
For the $A$-series the Weyl group is the symmetric group $S_n$. For the $B$ and $C$ series the Weyl group is the hyperoctahedral group $\mathbb Z_2 \wr S_n$.
A) Does the $D$-series Weyl group $S_n \...
- 421
2
votes
0
answers
86
views
Rewriting systems for finite groups [closed]
This is a question about rewriting systems & languages for finite groups. I'm sure everything must be in the literature somewhere, but I find it hard to navigate the references I have (for example ...
- 2,287
1
vote
0
answers
61
views
Finitely generated group in which maximal subgroups are commensurable
Is there a finitely generated torsion-free simple group such that every pair of maximal subgroups of the group are commensurable, i.e. for every pair of maximal subgroups $M$ and $N$ of the group, $|M:...
- 379
14
votes
2
answers
725
views
Are there any non-conjugation "extendible automorphisms" in the category of finite groups?
Let $\mathbf{Grp}$ be the category of groups. Given a subcategory $\mathscr{G}$ of $\mathbf{Grp}$ and $G\in\mathit{Ob}(\mathscr{G})$, a $\mathscr{G}$-extendible map on $G$ will here mean an assignment ...
- 21.1k
3
votes
1
answer
159
views
Does the fundamental group of a compact 3-manifold induce full profinite topology on the fundamental group of its boundary?
Let $M^3$ be a compact, orientable, irreducible 3-manifold with incompressible boundary. Let $S\subseteq\partial M$ be one of its boundary components. Does $\pi_1(M)$ induce the full profinite ...
- 605
15
votes
6
answers
669
views
Why, conceptually, does the torus normalizer in $G_2$ split?
Background: Let $G$ be a complex reductive group, $T$ a maximal torus, $N$ the normalizer of $T$ in $G$, and $W = N/T$ the Weyl group. All in all, we have a group extension
$$ 1 \to T \to N \to W \to ...
- 815
4
votes
1
answer
161
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Groups (not?) quasi-retracting onto $\mathbb{Z}$ via closest points projection
Inspired by this question we ask:
Suppose that $G$ is an infinite group. Suppose that $X$ is a finite generating set of $G$. Let $\Gamma = \Gamma(G, X)$ be the resulting Cayley graph. Does $\Gamma$ ...
- 28.1k
5
votes
1
answer
138
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Groups (not) quasi-retracting onto $\mathbb{Z}$
Let $X$ and $Y$ be metric spaces, with metrics $d_X$ and $d_Y$ respectively. A function $f\colon X\to Y$ is coarse Lipschitz if there exist constants $C,D>0$ such that $d_Y(f(x),f(x'))\le C d_X(x,x'...
- 3,730
1
vote
1
answer
119
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Proving that compact Hausdorff groups are cofiltered limits of compact Lie groups
What is the easiest way to show that a compact hausdorff topological group is a closed subgroup of a product of finite dimensional Lie groups?
Here are the relevant definitions:
Definition: (compact ...
user531936
3
votes
1
answer
106
views
Subgroups of a Weyl group fixing some vectors and its cohomology: MAGMA
I am trying to calculate the number of subgroups of the Weyl group $W(E_N)$ that fix certain vectors $L_i (i = 1,2,3)$ using Magma.
However, the output of the following code (especially #nicesubs) ...
- 1,364
8
votes
2
answers
352
views
Embedding f.g. groups in 2-generated groups
Let $G$ be a finitely generated group. Can $G$ be embedded as a finite-index subgroup of a 2-generated group? 100-generated?
I strongly doubt it but I don't know a counterexample.
- 9,617
14
votes
2
answers
1k
views
If every definable class admits a group structure, must global choice hold?
It is a remarkable fact, due to Hajnal and Kertész and explained very well in this MathOverflow answer by user Ashutosh, that the axiom of choice is equivalent to the assertion that every nonempty set ...
- 236k
1
vote
0
answers
172
views
Isomorphism classes of finite $\mathbb{N}$-groups
Where can I find resources on isomorphism classes of finite $\mathbb{N}$-groups, i.e. groups acted on by the monoid $(\mathbb{N}, +)$?
I edited this question to be more focused on what I'm interested ...
- 591
11
votes
1
answer
330
views
What is the minimal genus of a surface acted on by the symmetric group $S_n$?
For $G$ a finite group, it is easy to construct a (connected, orientable) surface with a faithful action of $G$. E.g.: take a disjoint union of $G$ many spheres, and add a 1-handle for every edge in ...
- 43.2k
1
vote
1
answer
209
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A question about automorphism group of abelian group
Does anyone know any references that describe automorphism group $\operatorname{Aut}(\mathbb R^n\times \mathbb T^m)$? I searched for a long time but couldn't find it.
- 71
1
vote
0
answers
48
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Length of the product of two elements of the subregular two-sided cell in the affine Weyl group of type A
The affine Weyl group of type $A_n$ can be described as follows. It is the group of all permutations $\sigma: \mathbb Z \to \mathbb Z$ such that $\sigma(i+n)=\sigma(i)+n$ and $\sum_{i=1}^n (\sigma(i)-...
- 2,964
3
votes
1
answer
151
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$C'(\lambda)$ small cancellation for $\lambda < \frac{1}{6}$
I'm working with Gromov's density model of random groups, and a nice fact is that for a fixed density parameter $0 \leq d \leq 1$, a generic group in the density model satisfies the $C'(2d)$ small ...
- 495
4
votes
1
answer
426
views
Is $\mathbb Z$ prime in the class of abelian groups?
Let $B$, $C$, and $D$ be abelian groups such that $\mathbb Z\times B$ is isomorphic to $C\times D$.
Is there a group $E$ such that $C$ or $D$ is isomorphic to $\mathbb Z\times E$?
Reference: page 263 ...
- 1,644
5
votes
1
answer
434
views
Poincaré duality and Mayer–Vietoris sequence
In his article, Davis states that if a $n$-dimensional Poincaré duality group $G$ splits along a subgroup $C$ then the cohomological dimension of $C$ must be $n-1$. I am struggling to understand why ...
- 73
1
vote
1
answer
115
views
Block-diagonal embedding of $U(n)$ into $U(mn)$
What is known about the subgroup $U(n)\subset U(mn)$ for $m,n\in\mathbb{N}$ given by the diagonal embedding
$$ \alpha\mapsto \text{diag}(\alpha,\cdots, \alpha),$$
for $\alpha$ appearing $m$ times?
For ...
2
votes
0
answers
99
views
Cohomological characterization of when $f: \pi_1(\Sigma_g) \to P$ factors through $F_g$ when $P$ is perfect
In previous questions on this site such as this one, it has been asked when a map $\varphi \colon G \to H$ of finitely generated groups factors through a free quotient meaning that there exists a ...
- 71
5
votes
2
answers
155
views
Explicit $2$-cocycle for $2^{1+2n}_+$
Let $p$ be a prime and $n\geq 1$, it is known that there are exactly two extraspecial groups of order $p^{1+2n}$, denoted by $p^{1+2n}_+$ and $p^{1+2n}_-$. They fit into central extensions
$$0\to(\...
- 245
0
votes
0
answers
42
views
Which finite group schemes over positive characteristic have projectivity detected by a set of (generalized) cyclic subgroups?
A finite algebra $B$ over a field $K$ has projectivity detected by a set $S = \{f_i : A \to B \mid i \in I\}$ of maps into $B$ from a fixed algebra $A$ if for any finite module $M$, $M$ is projective ...
- 532