All Questions
Tagged with reference-request gr.group-theory
700 questions
9
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3
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Where was it first stated that there are no 4-transitive finite groups other than symmetric, alternating and Mathieu groups?
It seems to be well-known that the six-transitive finite groups are the symmetric and alternating groups, and the only other four-transitive finite groups are the Mathieu groups (the statement can be ...
- 96.4k
9
votes
2
answers
2k
views
alternating and symmetric powers of the standard representation of the symmetric group
Let $n \geq 7$ and $V = \mathbb{C}^n$ be the standard representation for $S_{n+1}$, the symmetric group of cardinal $(n+1)!$
Let $k$ be an integer such that $2 \leq k \leq n$. Is it true or false ...
- 7,300
9
votes
1
answer
508
views
When is the augmentation ideal projective as RG-module?
Let $G$ be a finite group and let $R$ be a commutative ring.
I'd like to ask, if there is a theorem of the following kind:
The augmentation ideal $I_G$ is projective as RG-module, if and only if ... ?...
- 1,755
9
votes
1
answer
2k
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Finite groups in which all proper subgroups are cyclic
Is there any classification of finite group in which all proper subgroups are cyclic?
Would you please tell me a reference?
- 91
9
votes
3
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435
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How do small central extensions drop the dimension of a faithful representation?
Apologies in advance that this is a very soft question. I might be talking complete nonsense. But I hope I am talking about something that has even been studied...
I am interested in the phenomenon ...
- 2,851
9
votes
1
answer
738
views
Gromov hyperbolic groups which are solvable are elementary
I have read on wikipedia that a Gromov hyperbolic group which is solvable is elementary (i.e. virtually cyclic). Where can I find a proof of this fact?
There is a proof of a similar fact in Bridson-...
- 291
9
votes
1
answer
3k
views
Automorphism group of a finite group
I would like to ask if there exists an explicit description of $\mathrm{Aut}(G)$, the group of automorphisms of a finite group $G$, in particular, when $G$ is abelian. E.g., if $G = \mathbb{Z}/m\...
- 675
9
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3
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675
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Group extensions and actions on categories
Let G and H be two groups. There is a one-to-one correspondence between:
(i) an (isomorphism class of) extension of G by H, i.e. an exact sequence of group morphisms $1\to H\to E\to G\to 1$;
(ii) an ...
- 93
9
votes
1
answer
304
views
About the normal subgroups of Burnside groups
I was reading "On periodic groups of odd period $n\ge 1003$" of V. S. Atabekyan. He found that the Burnside group $B_n$ with $n\ge 1003$ has uncountably many normal subgroups. However, I was ...
- 91
9
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2
answers
762
views
Solutions of $x^d=1$ in the symmetric group
L Moser and M Wyman, On solutions of $x^d = 1$ in symmetric groups, Canad. J. Math., 7 (1955), pages 159-168, explored asymptotic behavior of the cardinality of such permutations:
$$f_d(n):=\#\{\pi\in\...
- 43.2k
9
votes
1
answer
573
views
$SL_2(\mathbf{Z},8\mathbf{Z})$ differs from $E_2(\mathbf{Z},8\mathbf{Z})$. Has this result appeared in the literature?
I know a proof that the congruence subgroup $SL_2(\mathbf{Z},8\mathbf{Z})$ differs from its subgroup $E_2(\mathbf{Z},8\mathbf{Z})$, but can't find this fact in the literature. Does anyone know a ...
- 127
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1
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1k
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Ping Pong and Free Group Factors
This question concerns alternative characterizations of free group factors. The ping pong lemma is a well-known criteria for the freeness of a group. I've often wondered if there is a ping pong like ...
- 7,057
9
votes
1
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893
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Kaplansky conjecture (consequences)
The Kaplansky conjecture says that: for any field $F$ and any torsion free group $G$, the group ring $F[G]$ does not have nontrivial idempotent elements.
Questions
Do we assume that $F$ has any ...
- 641
9
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2
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674
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Powers of finite simple groups
I have heard about the following result: for each finite simple non-abelian group $S$ and each natural number $r\ge 2$ there exists a number $n=n(r,S)$ such that the power $S^n$ is $r$-generator but $...
- 331
9
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1
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290
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Calculations of nonabelian group cohomology of R^n
I am looking at $H^1(\mathbb{R}^n,G)$ where $G$ is a finite 2-group. I'm wondering if such things have been calculated. I'm afraid I can't say I know anything here, past the result that this ...
- 35.5k
9
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1
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1k
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Easy argument for "connected simple real rank zero Lie groups are compact"?
Let $G$ be a connected simple Lie group. It is known that if $G$ has real rank zero, then $G$ is compact.
Background: every connected (semi)simple Lie group $G$ (with Lie algebra $\mathfrak{g}$) has ...
- 223
9
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1
answer
735
views
Where has this structure been observed?
$\newcommand{\M}{\mathcal{M}}$Let $M$ be a monoid. Consider the following structure:
$R_X,R_Y:\mathbb{Z}^2 \to M$ satisfying the following "compatiblity-relation":
$$R_X (x, y) \cdot R_Y (x +...
- 6,741
9
votes
1
answer
271
views
Original references for the Hall - Witt identity
The group identity
$$
[[a,b^{-1}],c]^b \cdot [[b,c^{-1}],a]^c \cdot [[c,a^{-1}],b]^a = 1
$$
is commonly attributed to Hall and Witt (here $x^y:=y^{-1}xy$ and $[x,y]:=x^{-1}y^{-1}xy$). However, ...
- 17k
9
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1
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458
views
Fuchsian groups and Eichler's result
Let $G$ be a Fuchsian group of first kind contained in $\text{PSL}_2(\mathbb{R})$. A result of Eichler says, there exists a finite set $S\subset G$ such that any $\gamma$ in $G$ can be written as a ...
- 521
9
votes
2
answers
571
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Algorithm for group cohomology
Let $G$ be a finite group, and let $0\to M_1\xrightarrow{\iota} M_2\xrightarrow{\pi} M_3\to 0$ be a short exact sequence of $G$-modules (finitely generated over $\mathbb Z$, not necessarily free). I ...
- 161
9
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1
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464
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Branching Rule for alternating groups
Let $A_n$ be the alternating group of degree $n$. What is the branching rule for the subgroup $A_{n-1}\subset A_n$, i.e., the structure of the restriction of ordinary irreducible representations of $...
- 377
9
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1
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485
views
A residually finite modification of the wreath product
I have been looking for ways to construct examples of finitely generated residually finite groups that are poly-(locally virtually abelian) but not virtually solvable.
If $K$ is a finite non-solvable ...
- 740
9
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2
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485
views
Reference for restriction of a simple module over a splitting field to a smaller field?
This is mainly a request for a straightforward reference (preferably at textbook level). The question comes up while responding to a question raised by non-specialists in finite group representations....
- 52.9k
9
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1
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409
views
The maximum order of torsion elements in ${\rm GL}_n(\mathbb{Z}_p)$ or ${\rm GL}_n(\mathbb{F}_p[[T]])$
This question is inspired by Upper bound on order of finite subgroups of GL_n(Z_p)?. It's showed that the supremum of orders of finite subgroups of ${\rm GL}_n(\mathbb{Z}_p)$ is finite and can be ...
- 863
9
votes
3
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548
views
Spectral radius of a finitely generated group
Let $G$ be a finitely generated group and $\Gamma$ be its Cayley graph with the usual word metric. Let $\mu$ be a symmetric non-degenerate measure on $G$ (maybe with finite support or smooth), and ...
- 1,407
9
votes
2
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772
views
Characters of orthogonal groups as symmetric functions
This question was asked on MSE some time ago, here, but got no attention.
The Schur functions are characters of irreps of the unitary group, $s_\lambda(U)=Tr(R_\lambda(U))$. They are symmetric ...
- 2,552
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1
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337
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amenable + without $BS(m,n)$+finite $K(G,1)$implies virtually cyclic?
I heard from someone that the following problem is an open question.
(Open Problem 1)For a countable discrete group $G$, suppose it does not contain any Baumslag-Solitar subgroups $BS(m,n):=\langle x,...
- 1,528
9
votes
2
answers
701
views
Reference for embeddings of reflection groups (related to folding ADE Coxeter graphs)?
There are a couple of indirect methods, using Lie theory or Springer's general theory of regular elements in (real, complex) reflection groups, to construct natural embeddings among certain Weyl ...
- 52.9k
9
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2
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1k
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Is it known if the absolute Galois group is "divisible"?
The definitions of a divisible group that I have seen all seem to assume abelian is an a priori property of the group. My question is as to whether or not it is known that--given a non-torsion element ...
- 1,049
9
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1
answer
1k
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What is the outer automorphism group of $\operatorname{SL}(2,\mathbb{F}_q)$?
I'm looking for a reference for a description of the outer automorphism groups of $\operatorname{SL}(2,\mathbb{F}_q)$ for $q = p^n$.
I'm sure such a thing must exist somewhere, but I'm having trouble ...
- 7,527
9
votes
1
answer
311
views
Reference for Schur multiplier identity
Let $G$ be a finite group and $H$ a normal subgroup of $G$. I recently stumbled upon the following identity for the Schur multiplier of $G/H$:
$$\operatorname{H}_2(G/H,\mathbb{Z}) \cong \frac{\...
- 93
9
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1
answer
460
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Connections between linear representations and permutation representations
A finite group $\Gamma$ might be represented by a linear transformation
$$\rho : \Gamma\to\mathrm{GL}(\Bbb R^d),$$
or by permutations
$$\phi :\Gamma\to\mathrm{Sym}(n).$$
Of course, latter ones can ...
- 13.6k
9
votes
1
answer
384
views
Reference for the fact that $SL_n(O_K)$ surjects onto $SL_n(O_K/I)$ for any ideal I
Let $\mathcal{O}_K$ be the ring of integers in an algebraic number field $K$ and let $I \subset \mathcal{O}_K$ be a nonzero proper ideal. It is not hard to see that the map $\text{SL}_n(\mathcal{O}_K)...
- 91
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1
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410
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On a result by Rubin on elementary equivalence of homeomorphism groups and homeomorphisms of the underlying spaces
In the known paper On the reconstruction of topological spaces from their group of homeomorphisms by Matatyahu Rubin several deep reconstruction theorems of the form "if $X$ and $Y$ are ...
- 3,011
9
votes
1
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495
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Divergence of Groups and Metric Spaces
Several papers, including this and this claim that divergence of finitely generated groups and metric spaces have been introduced by Misha Gromov in his paper "Asymptotic invariants of infinite groups"...
user6976
9
votes
1
answer
223
views
$p$-adic analytic pro-$p$ group satisfies a pro-$p$ identity?
Let $p$ be a prime. Let $w$ be an element of a free pro-$p$ group $F_r$ of finite rank $r\geq 2$. Then we say that a pro-$p$ group $G$ satisfies the pro-$p$ identity
$w$ if for every homomorphism $ f:...
- 667
9
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1
answer
309
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Comparing cohomology of a total complex with the cohomology of semidirect product
$\DeclareMathOperator{\Tot}{Tot}$I have the following problem. Let $H$ and $G$ be groups such that $H$ acts on $G$, i.e., there exists a group homomorphism $H\to \mathrm{Aut}(G)$ and let $M$ be an ...
- 1,759
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1
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435
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Questions on the group $\mathrm{GL}(H)$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\U{U}$Let $H$ be an infinite dimensional complex Hilbert space. Consider the group $\GL(H)$ of bounded invertible operators on $H$.
Question 1. I've ...
- 1,586
9
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1
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230
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Yang-Mills algebra and lower central series of surface groups
Here is a connection that I recently noticed, but I haven't quite been able to make sense of. It might follow from well-known facts; apologies, if so. This is quite far from my area.
First, in "...
9
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0
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254
views
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 ...
- 2,306
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0
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275
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Is “simplicity is elementary” still hard? (Felgner’s 1990 theorem on simple groups, and subsequent work)
I came across a reference in this MathOverflow answer to an intriguing result of Ulrich Felgner [1]: among finite non-Abelian groups, the property of being simple is first-order definable. According ...
- 21.3k
9
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0
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456
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Hopficity of Baumslag-Solitar groups
I am struggling to find the exact source (with proofs) of the following ''well-known'' statement:
the Baumslag-Solitar group $BS(m,n)=\langle a,t \mid ta^m t^{-1}=a^n\rangle$ is Hopfian if and only if ...
- 3,181
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0
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254
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Decomposition of linear groups into free products
I recently learnt about Nagao's theorem which states $SL_2(k[t])\cong SL_2(k)\ast_{B(k)}B(k[t])$ for a field $k$. I read in "A.W. Mason, Serre's generalization of Nagao's theorem" that a theorem of ...
- 5,901
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0
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230
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Using Property (T) to approximate invertible matrices
In the wikipedia article for Kazhdan's Property (T), there's an intriguing application:
Similarly, groups with property (T) can be used to construct finite sets of invertible matrices which can ...
9
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0
answers
329
views
'Infinitesimal' elements of a topological group
Let $G$ be a topological group, and let $M$ be the intersection of all conjugacy-invariant neighbourhoods of the identity in $G$ (in other words, the set of elements that can be taken arbitarily close ...
- 4,728
8
votes
4
answers
659
views
Normal Covering of a Finite Group
Suppose $G$ is a finite group and $N_1, N_2, \cdots, N_k$ are proper normal subgroups of $G$. The set $\{ N_1, \cdots, N_k\}$ is called a normal cover for $G$, if $G = \cup_{i=1}^kN_i$. I need to the ...
8
votes
4
answers
1k
views
Degree of commutativity of finite groups and subgroups
Recently I started reading some articles about the
degree of commutativity of finite groups. I have some questions:
In "Subgroup commutativity degrees of finite groups" Tarnauceanu
proposes ...
user11178
8
votes
2
answers
617
views
Relative/acylindrical hyperbolicity of free-by-cyclic groups
Is this statement true?
Let $\mathbb{F}$ denotes a finitely generated free group, $\Phi$ an automorphism of $\mathbb{F}$ and $\varphi$ its image in $\mathrm{Out}(\mathbb{F})$.
If $\varphi$ is ...
- 1,755
8
votes
3
answers
559
views
Reference for tetrahedral Coxeter group
Let $G$ be the group with 4 generators, each of order 2, such that the product of any 2, say $ab$, has order 3 (i.e., $ababab=e$).
That is, this is an infinite reflection group with Coxeter diagram a ...
- 83
8
votes
1
answer
396
views
Which group do two generic $2\times 2$ triangular matrices generate?
Let $A,B$ be two generic (in particular invertible) $2\times 2$ upper-triangular complex matrices. They generate a countable group $G$, the commutator subgroup of $G$ is abelian. Are there other ...
- 109k