Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
7,823
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$p$-modular splitting systems and the characteristic of the ring $\mathcal{O}$
Let $k=\overline{k}$ be a field of characteristic $p$.
Let $(K,\mathcal{O},k)$ be a $p$-modular system.
Let both $k$ and $K$ be splitting fields for $G$ and its subgroups.
The ring $\mathcal{O}$ can ...
- 311
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Extension of base field for modules of groups and cohomology [duplicate]
Let $G$ be a group and let $K/k$ be a field extension. Suppose that $V$ is a $kG$-module, and let $V_K = K \otimes_k V$ be the $KG$-module given by changing the base field.
Is it true that $H^n(G,V_K) ...
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2
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Why does the monster group exist?
Recently, I was watching an interview that John Conway did with Numberphile. By the end of the video, Brady Haran asked John:
If you were to come back a hundred years after your death, what problem ...
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0
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112
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Idempotent conjecture and non-abelian solenoid
Is there a discrete non-abelian group whose dual in a reasonable sense is isomorphic to the solenoid constructed via a sequence of quaternions $S^3$ instead of a sequence of circles? The motivation ...
- 280
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94
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Idempotent conjecture and (weak) connectivity of (a reasonable) dual group
What is an example of a torsion free discrete abelian group $G$ whose dual space $\hat{G}$ is not a path connected space?
The Motivation: The motivation comes from the idempotent conjecture of ...
- 280
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178
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What is the latest progress on the Andrews-Curtis Conjecture?
Out of curiosity . . .
What is the latest progress on the Andrews-Curtis Conjecture?
What's available online seems limited. (See the Wikipedia article linked to above.)
I found the following here:
...
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6
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1
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202
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Amalgamation of finitely generated finite exponent groups
Suppose $C\leq A,B$ are finitely generated groups of finite exponent. Can $A$ and $B$ be amalgamated over $C$ in a group of finite exponent? What about if $A,B$ are periodic, can we find a periodic ...
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101
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Finite groups of prime power order containing an abelian maximal subgroup
Let $G$ be a finite $p$-group containing an abelian maximal subgroup. Then it is a well-known result that $|G:Z(G)|=p|G'|$. If in addition $G$ is of nilpotent class 2, then $|G:Z(G)|\leq p^{r+1}$, ...
0
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1
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124
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A question on irreducible affine Coxeter groups
I have a question about affine Coxeter groups when reading Humphreys's book:
Let $(W,S)$ be an irreducible affine Coxeter group, $M=(m_{ij})$ be its Coxeter matrix, and $\{\alpha_s\}_{s\in S}\in V$ be ...
- 435
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Smith normal form and last invariant factor of certain matrices
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts can provide some relevant insight.
Suppose we have row vectors $x_1$, $x_2$ , $y_1$ , $y_2 ...
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Mackey coset decomposition formula
I have a question about following argument I found
in these notes on Mackey functors:
(2.1) LEMMA. (page 6) Let $G$ be a finite group and $J$ any subgroup. Whenever $H$ and $K$ are subgroups of $J$, ...
- 5,716
13
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2
answers
725
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Is it possible for a direct product to be isomorphic to the Zappa–Szép product?
Let $A$ and $G$ be two groups. Let $\alpha:G\rightarrow\operatorname{Bij}(A)$ be a group homomorphism and $\beta: A\rightarrow\operatorname{Bij}(G)$ an anti-homomorphism satisfying some conditions ...
- 383
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115
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normalizer info for subgroups
In [1], Griess classified the maximal nontoral elementary abelian subgroups of algebraic groups. For the exceptional types, normalizer info was also given. Is there any work out there providing ...
- 501
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2
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349
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Twisted forms with real points of a real Grassmannian
Let $X={\rm Gr}_{n,k,{\Bbb R}}$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb R}^n$.
We regard $X$ as an ${\Bbb R}$-variety with the set of complex points $X({\Bbb C})={\rm Gr}_{n,k,{\...
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Is there a maximum length of the chief series of $SL(n,q)$?
Motivation:
I'm studying certain properties of conjugation in $SL(n,q)$. There's a nice number, a bit like a covering number, that one can associate with an arbitrary group. In writing a programme in ...
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404
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Finite conjugacy classes
Let $G$ be an infinite group. Let $N_0$ be the set of all $x\in G$ for which the conjugacy class $\{y^{-1}xy: y\in G\}$ is a finite set. Clearly $N_0$ is a normal subgroup. Iteratively, form an ...
- 2,118
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0
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62
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Are the integer points of a simple linear algebraic group 2-generated?
Set Up:
Let $ K $ be a totally real number field. Let $ \mathcal{O}_K $ be the ring of integers of $ K $. Let $ G $ be a simple linear algebraic group. Suppose that $ G(\mathbb{R}) $ is a compact Lie ...
- 3,429
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1
answer
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Is the infinite product of solvable groups amenable?
I am interested in the amenability properties of infinite products of solvable groups. The following facts are well-known:
Any solvable group is amenable.
The class of solvable groups is closed under ...
- 153
4
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1
answer
191
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Projective representations of a finite abelian group
Projective representations of a group $G$ are classified by the second group cohomology $H^2(G,U(1))$. If $G$ is finite and abelian, it is isomorphic to the direct product of cyclic groups
$$
G\cong ...
- 839
6
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2
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Presentation of the fundamental group of a complex variety
Let $X$ be a connected smooth complex algebraic variety and $Z=\bigcup_{i=1}^r Z_i$ be a union of smooth connected hypersurfaces, satisfying that each two intersect transversally. Assume for ...
14
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2
answers
548
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Existence of a regular semisimple element over $\mathbb{F}_{q}$
This is probably old, a Chevalley level of old, but I'm not at all an expert in this field so I need help.
Let $G$ be a simply connected (almost) simple linear algebraic group defined over $K=\mathbb{...
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169
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Cyclic numbers of the form $2^n + 1$
A cyclic number (or cyclic order) is a number $m$ such that the only group of order $m$ is the cyclic group $\mathbb{Z}/m\mathbb{Z}$. The set of cyclic numbers admits a couple of cute number-theoretic ...
- 333
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104
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Homology functors and weak cofibers
I'm looking at a remark in the paper
Kainen, Paul C., "Weak Adjoint Functors", Mathematische Zeitschrift 122 (1971).
It is supposed to prove that generalized homology functors fail to ...
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4
votes
1
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268
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Understanding $(\mathbb{Z}/3)^2 \times_{\mathbb{Z}/3} M$
I'm currently reading "Bordism of Elementary Abelian Groups via Inessential Brown-Peterson Homology" by Hanke (arXiv:1503.04563) and have come across some notation that I'm not familiar with....
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1
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280
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Maximal subgroups of $S_{\Bbb N}$ and $A_{\Bbb N}$
Let $S_{\Bbb N}$ be the countable symmetric group of all permutations of the naturals with finite support and $A_{\Bbb N}$ --- the corresponding alternating group.
How to describe all maximal ...
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1
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228
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Maximal subgroup in $S_{10}$
Consider the set of unordered pairs $\{(i,j)\}$, $i<j, i=1,2, \ldots, 2k+1$, $j=i+1, \dots, 2k+2$, and the group $G=S_{k(2k+1)}$ of all permutations of those pairs.
Is the subgroup of the ...
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Is solvability semi-decidable?
Let $G = \langle A \mid R \rangle$ be a finitely presented group, given by a finite presentation. If $G$ is abelian, then we can verify this fact: simply verify the fact that $[a, b] = 1$ for all ...
4
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1
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Action of braid groups on regular trees
Question:
Are there any well known actions of braid groups on trees? For example is there some action of a braid group $ B_n $ on a $ p $ regular tree for some $ p $ such that the action is transitive ...
- 3,429
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1
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388
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Classes of groups with polynomial time isomorphism problem
It is known that the isomorphism problem for finitely presented groups is in general undecidable. What are some classes of groups whose isomorphism problem is known to be solvable in polynomial time? (...
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212
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What can lattices tell us about lattices?
A general group-theoretic lattice is usually defined as something like
A discrete subgroup $\Gamma$ of a locally compact group $G$ is a lattice if the quotient $G/\Gamma$ carries a $G$-invariant ...
- 1,729
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2
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Subsets of free groups contained in $2$-generated subgroups
$\DeclareMathOperator\rank{rank}$Let $F$ be a non-cyclic free group.
For which finitely generated subgroups $H< F$ such that $H$ is not of finite index in a free factor of $F$ does there exist a ...
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Does a finitely presented amenable group contain a unique maximal finite normal subgroup?
O. Baues and F. Gruewald in their paper "AUTOMORPHISM GROUPS OF POLYCYCLIC-BY-FINITE GROUPS AND ARITHMETIC GROUPS" have stated that a polycyclic-by-finite group has a unique maximal finite ...
- 1,172
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2
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The orbits of an algebraic action of a semidirect product of a unipotent group and a compact group are closed?
We consider real algebraic groups and real algebraic varieties. It is known that the orbits of an algebraic action of a unipotent algebraic group $U$ on an affine variety are closed. The orbits of an ...
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2
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440
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When are finite-dimensional representations on Hilbert spaces completely reducible?
Let $G$ be a group and $\pi$ be a finite-dimensional (not necessarily unitary) representation of $G$ on a complex Hilbert space $H$. We shall say that $\pi$ is completely reducible if there exists a ...
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Condition on $q$ for inclusion $p^{1+2r}\cdot\operatorname{Sp}(2r,p)\leqslant \operatorname{GU}(p^r,q)$
Let $p$ be an odd prime. What's the condition on $q$ for
$$
p^{1+2r}\cdot\operatorname{Sp}(2r,p)\leqslant \operatorname{GU}(p^r,q)\;?
$$ I did some computation and seemed that $q\equiv -1$(mod $p$) ...
- 501
16
votes
1
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726
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A "simpler" description of the automorphism group of the lamplighter group
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts can point me to some relevant references.
The lamplighter group is defined by the ...
- 763
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votes
0
answers
115
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group generated by unipotents in arithmetic subgroup is finitely generated
Let $G$ be a semisimple algebraic $\mathbb{Q}$-group and $\Gamma$ an arithmetic subgroup of $G$. In particular $\Gamma$ is finitely generated.
Denote by $\Gamma^{u}$ the set of unipotent elements in $\...
4
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1
answer
118
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CFSG-free proof for classifying simple $K_3$-group
Let $G$ be a finite nonabelian simple group.
We call $G$ a $K_3$-group if $|G|=p^aq^br^c$ where $p,q,r$ are distinct primes and $a,b,c$ are positive integers.
My question is: Is there a CFSG-free ...
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Morse theory on outer space via the lengths of finitely many conjugacy classes
Let $F_n$ be the free group on letters $\{x_1,\ldots,x_n\}$ and let $X_n$ be the (reduced) outer space of rank $n$. Points of $X_n$ thus correspond to pairs $(G,\mu)$, where $G$ is a finite connected ...
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Proof of Zimmer's cocycle super-rigidity theorem
I was reading the proof of Zimmer's cocycle super-rigidity theorem from the book 'Ergodic theory and semi-simple groups' by Robert Zimmer (Theorem 5.2.5, page 98). But I am not able to understand it. ...
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0
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90
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Constructing tensor structures for representations over group schemes
Let $A$ be an algebra over a field $k$. Let's say a tensor structure for modules over $A$ is any functorial assignment of an $A$-module structure to $M\otimes_kM'$ for $A$-modules $M, M'$. A good way ...
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1
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Profinite groups with isomorphic proper, dense subgroups are isomorphic
I am developing a sort of standard representation for profinite quandles. This involves profinite groups a lot, actually. In one part of my construction the filtered diagram used to construct a ...
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2
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585
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Involutions in the absolute Galois group (and the Axiom of Choice)
It is known that the only elementary abelian $2$-groups (finite and nonfinite) in $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ are in fact finite and cyclic – that is to say, they are of order $2$....
- 4,025
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1
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168
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$\mathrm{PSL}_3(4)$ inside the Monster group
Which quasisimple groups with central quotient $G\cong\mathrm{PSL}_3(4)$ are isomorphic to subgroups of the Monster sporadic group? So far I know that $G$ itself is not and that $2\cdot G$, $2^2\cdot ...
- 2,613
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votes
0
answers
64
views
Are Gromov-hyperbolic groups roughly starlike? [duplicate]
Given a Cayley graph of a finitely generated Gromov-hyperbolic group $G$, does there exists $R>0$ such that every element $g \in G$ is at most distance $R$ away from a geodesic ray starting at ...
- 61
2
votes
1
answer
55
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Can the stabiliser of a 'parabolic end' of a group stabilise an invariant line?
Let $G$ be a group acting freely and cocompactly on an infinite-ended graph $\Gamma$. In particular, $G$ is finitely generated and acts as a convergence group on the Cantor set $\rm Ends(\Gamma)$.
Let ...
- 537
2
votes
1
answer
184
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Sparsity of q in groups PSL(2,q) that are K_4-simple
One of the problems that has come up during my research concerns $K_4$-simple groups (simple groups with $4$ prime divisors). The only (potentially) infinite family of groups satisfying this condition ...
- 23
13
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0
answers
192
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Are there free and discrete subgroups of $\mathrm{SL}_2(\mathbb{R}) \times \mathrm{SL}_2(\mathbb{R})$ that are not Schottky on any factor?
$\DeclareMathOperator\SL{SL}$Does there exist a free and discrete subgroup $\Gamma < \SL_2(\mathbb{R}) \times \SL_2(\mathbb{R})$ such that neither $\pi_1(\Gamma)$ nor $\pi_2(\Gamma)$ is free and ...
- 1,354
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votes
0
answers
40
views
Semigroups containing an appropriate subgroup
We are looking for a class of non-monoid semigroups $S$ (resp. monoids $M$) satisfying the following conditions:
(1) $S$ has a left identity;
(2) There exists a subgroup $H$ of $S$ (resp. $M$) such ...
- 985
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votes
2
answers
208
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Must an isomorphism preserving graph transformation preserve the order of the automorphism group?
Let $F$ be some function graph to graph which preserve graph isomorphism.
Example of such $F$ are the line graph, the $k$-subdivision of $G$
and many others.
$F$ need not preserve the order, the ...
- 24.1k