All Questions
Tagged with reference-request gr.group-theory
700 questions
2
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Does $G\times H$ have a dual when $G$ and $H$ have?
Let $G$ and $H$ be two groups with duals. Does $G\times H$ have a dual?
A group $G$ has a dual iff the lattice of its subgroups is order-isomorphic to the dual of the subgroup lattice of some other ...
- 1,145
2
votes
0
answers
417
views
An equivariant Hahn embedding theorem?
The Hahn Embedding Theorem asserts that for any (linearly) ordered abelian group $\Lambda$, there exists a linearly ordered indexing set $\Omega$ such that $\Lambda$ admits an order-preserving group ...
- 1,089
2
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0
answers
165
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Reference request: injective homomorphisms between unitary groups
Let $U(n)$ be the group of unitary $n\times n$ matrices over $\mathbb{C}$. Is there a classification of the continuous, injective group homomorphisms $U(m)\to U(n)$? If so, is there a modern account ...
- 1,336
2
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0
answers
909
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List of commutator identities and equivalences
Let $G$ be a group and let $[a,b]=a^{-1}b^{-1}ab$ be the commutator of $a$ and $b$ in $G$. There are several well-known commutator identities such as
$[x, z y] = [x, y]\cdot [x, z]^y$
and
$[[x, y^{-...
- 1,108
2
votes
0
answers
281
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Chapter 28 of Berkovich, Zhmud, Characters of finite groups. Part 2
The MathSciNet review of the book Berkovich, Zhmud, Characters of finite groups. Part 2, says the following:
...Let $k(G)$ be the number of conjugacy classes of the group $G$, $T(G)$ the sum of the ...
- 761
2
votes
0
answers
153
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Reference request for a result on subsets unlikely to be hit by random walks in a group
Suppose we are performing a random walk in a group. More precisely, we have a finite generating set $S$ of a group $G$ and the probability of walking along generator $s$ is given by $\mu(s)$ for some ...
- 21
1
vote
2
answers
557
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Is there formula name and proof for this theorem ? [closed]
The formula answers: how many tuples $(\sigma_1,\sigma_2,...,\sigma_n)$ of elements of a given group G such that
(1) $\sigma_i\in C_i$ , where $C_i$ stands for conjugacy class.
(2) $\sigma_1\...
- 11
1
vote
2
answers
287
views
Faithfully flat modules over a group algebra
Suppose we have the following data:
1) A group ring $\mathbb{Z}[G]$, where $G$ is a torsion free group.
2) $M_{\bullet}$ a bounded (above and below) chain complex of $\mathbb{Z}[G]$-modules such ...
- 71
1
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1
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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
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2
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448
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Lattices in general totally disconnected locally compact groups
Besides automorphism groups of trees and buildings, I was wondering if the lattices in general totally disconnected locally compact groups have been studied in the literature? I appreciate if you ...
user23860
1
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1
answer
362
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Odd subgroup of $\mathrm{GL}(n,\mathbb{Z})$
The group $\mathrm{GL}(n,\mathbb{Z})$ acts on $(\mathbb{Z}/2\mathbb{Z})^n$ by right multiplication (the same kind of things can be done with left action). I denote by $H\subset \mathrm{GL}(n,\mathbb{Z}...
- 7,722
1
vote
3
answers
840
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An extension of Lagrange's theorem to semigroups?
The question is fairly dry: Is there any semigroup analogue of Lagrange's theorem for groups (counting as a generalization of the latter)? Let me guess the answer: Obviously yes. So the real question ...
- 10.5k
1
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2
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742
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Automorphism group of the affine groups AGL(n,q), ASL(n,q)
I have a question. The automorphism group of the linear groups $GL(n,q)$, the group of linear transformations of $V = \mathbb{F}_q^n$, and $SL(n,q)$, the subgroup of $GL(n,q)$ consisting of elements ...
- 731
1
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1
answer
804
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Classification of Special $p$-Groups
A finite $p$-group is said to be special if $Z(G)=G'$. Is there classification of special $p$-groups? (Please suggest references, if classification is done. If the classification is incomplete, please ...
- 113
1
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1
answer
346
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Reference on elements of finite order in principal congruence subgroups of symplectic groups
We should start with the definition of the symplectic group for an arbitrary ring $R$.
The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $M=J_g^t M J_g$ with $...
- 85
1
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1
answer
115
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Bounds for Khukhro-Makarenko theorems
Let’s define the set of outer-commutator group words $OC \subset F_\infty = F[x_0, x_1, …, x_n, …]$ using the following recurrence:
$$\forall i \in \mathbb{N} \text{ } x_i \in OC$$
$$\forall u, v \...
- 2,618
1
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2
answers
341
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Copies of ax+b inside the AN part of an Iwasawa decomposition?
As a relative novice to the structure theory of Lie algebras and Lie groups, the following is what I can gather from reading parts of Helgason's book DG, Lie groups and symmetric spaces and Knapp's ...
- 25.8k
1
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1
answer
301
views
Reference for a proof of the Dehn presentation
I would like a reference for a proof that the Dehn presentation is a presentation of the fundamental group of the knot complement in $\mathbb{S}^{3} $.
- 13
1
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1
answer
233
views
Transfer for the group of coinvariants: a reference request
Let $G$ be a group and $M$ be a $G$-module,
that is, an abelian group written additively on which $G$ acts:
$$ (g,m)\mapsto g m.$$
We consider the group of coinvariants
$$ M_G:=G/\langle g m -m\ |\ g\...
- 14.2k
1
vote
1
answer
214
views
Conditions of P for existence of orthogonal matrix Q and permutation matrix U satisfying QP = PU
Question: Let $P\in \mathbb{R}^{d\times n}$ be a $d$-rank real matrix and $PP^T = c I_d$ with a certain constant $c > 0$. Under what additional conditions of $P$ does there exist an orthogonal ...
- 187
1
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1
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259
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Which properties can be read off the balls of a Cayley graph?
For which properties (P) [of groups] does the following hold:
given a group $G$ which has a finite presentation with at most $n$ relations of length at most $\ell$, there is a $R(n,\ell)$ so that, if ...
- 4,432
1
vote
1
answer
150
views
Lemma for proof of Jordan-Hölder Theorem [closed]
Lemma
Let $G$ be a group, $K \triangleleft G$ a normal subgroup and $H_j \triangleleft H_i$ two subgroups of $G$ with $H_j$ a normal subgroup of $H_i$.
Then there is an isomorphism
$$(H_i K)/(H_j K)...
- 708
1
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1
answer
166
views
Reference for a proof of cancellation property of braid monoids
Let $M$ be a monoid. If $ab=ac$ implies that $b=c$, $a,b,c \in M$, then $M$ is said to have the left cancellation property. Similarly, the right cancellation property is $ba=ca$ implies that $b=c$.
...
- 6,201
1
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1
answer
195
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Reference request: The commensurator of an arithmetic lattice is a simple group
I am interested in a reference and proof for some version of the following (folklore?) statement:
``Let $G$ be a (semi)simple Lie group (with no compact factors and trivial centre) and let $\Gamma$ ...
- 855
1
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1
answer
312
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Topological Invariants for Group
Let $\mathbf{Grp}$ be the category of groups and $\mathbf{Top}$ be the category of topological spaces. To each group $(G, \circ_G)$, we can associate a topological space $(G,\tau_G)$ the basis for ...
user57432
1
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1
answer
339
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Type $C_n$ Weyl group contains in the centralizer of the longest word $w_0$ in $S_{2n}$
Are there some references about the proof of the following fact?
Type $C_n$ Weyl group lies in the centralizer of the longest word $w_0$ in $S_{2n}$.
Thank you very much.
- 6,201
1
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1
answer
107
views
Name of the class of linearly ordered groups with no minimal positive element
Is there a special name for a linearly ordered group $G$ such that for every positive element $g\in G$ there exists an element $h\in G$ such that $e<h<g$?
- 5,409
1
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1
answer
262
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Natural actions of quotients of automorphism groups
I've stumbled upon a construction which seems to be very much classical, and yet I found nothing definite about it so far in available sources. Let $\Lambda$ be a normal subgroup of the automorphism ...
- 303
1
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1
answer
578
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Fundamental inequality of entropy in random walks
I'm looking for a reference for an inequality related to the "fundamental inequality" about entropy and rate of escape of random walks (on the Cayley graph of a group). Namely,
$\textbf{Question}$: ...
- 4,432
1
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1
answer
353
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Decomposition of an induced representation
If there is a finite group $G$ with a cyclic normal subgroup $C_n$, one can describe the indecomposable representations of $G$ through induction. How does $Ind_{C_n}^G$ decompose? For representations ...
- 13
1
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1
answer
95
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Quotients of pro-$p$ groups linear over a complete Noetherian local ring
Let $R$ be complete Noetherian local ring with finite residue field $\mathbb{F}$ of characteristic $ p $. We say that a pro-$p$ group $G$ is linear over $R$ if it is isomorphic to a closed subgroup of ...
- 863
1
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1
answer
260
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A group-theoretic lemma in a paper by Ershov and He
In the proof of Lemma 2.1 in
Ershov, Mikhail; He, Sue, On finiteness properties of the Johnson filtrations, ZBL06904638,
the authors claim the following (without proof).
Let $G$ be a finitely ...
- 1,317
1
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1
answer
89
views
Element of order $p$ and finite height $\geq1$ in a reduced abelian group $p$-group with an element of order $p^2$
This is a reference request for the following statement:
Fact:
Let $G$ be a reduced abelian $p$-group with an element of order $p^2$. Then $G$ contains an element of order $p$ and of finite height at ...
- 343
1
vote
1
answer
435
views
Direct proof that free groups are sofic
I am looking for a reference (or a simple proof) of the fact that a free group is sofic. The preferred dynamical definition of a sofic group seems to be that
there is a sequence of finite sets $V_n$ ...
- 23.2k
1
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1
answer
255
views
Reference for the delooping of groups (or rings)
This nlab article defines the delooping groupoid of a group, but it does not provide a reference. I am using deloopings of rings, and I would like to refernce to a more classical source.
Could you ...
- 349
1
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0
answers
44
views
Lower bound for restricted sumset in ordered groups
Recently in The restricted sumsets in finite abelian groups it is proved that
Suppose that $k \geq 2$ and $A$ is a non-empty subset of a finite abelian
group $G$ with $|G| > 1$. Then the ...
1
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0
answers
89
views
The base group of a wreath product of an abelian group by $ {\mathbb{Z}}$ is a characterstic subgroup
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can direct me to some relevant results.
Let $A$ be a finitely generated abelian group,...
- 823
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0
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172
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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 ...
- 631
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0
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92
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The existence of such homomorphism [closed]
Are there any papers or books that investigate/discuss the relationship between conjugacy classes and normality for the existence of non-trivial homomrphism f:G->H were H is some nontrivial ...
- 61
1
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0
answers
71
views
Finitely presentable groups are residually finite if and only if they are universally pseudofinite
Suppose $G$ is finitely presentable. Then residual finiteness of $G$ is equivalent to $G$ satisfying the universal theory of finite groups (equivalently, to every existential statement true in $G$ ...
- 1,338
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0
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134
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Isomorphic quotients of a countably infinitely-generated free abelian group
Let $F$ denote the free abelian group on countably infinite generators. I am trying to understand the relationship between normal subgroups $A$ and $B$ of $F$ with isomorphic quotients. So is there a ...
- 11
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0
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188
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Does every amenable group $G$ admit a two-sided Folner sequence?
By two-sided Følner sequence I mean a sequence $(F_N)_N$ of subsets of $G$ which is both a left-Følner and a right-Følner sequence.
Context: I just came up with this question and surprisingly I haven'...
- 10.6k
1
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0
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110
views
Character table of $\mathrm{P\Gamma L}_2(q)$ with $q$ even
Let $q = 2^f$ for some integer $f\geqslant 3$. The character table of $\mathrm{SL}_2(q)\cong\mathrm{PSL}_2(q)$ can be deduced from the character table of $\mathrm{GL}_2(q)$ (see, for example, Exercise ...
- 379
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0
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83
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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
1
vote
0
answers
98
views
Existence of countable dense normal subgroups of global Galois group
Let $K$ be a number field, $S$ a finite set of primes of $K$ and $ G_{K,S} $ the Galois group of the maximal extension of $ K $ (inside a fixed algebraic closure of $K$) unramified outside $ S $. In ...
- 863
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0
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116
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List of automorphism groups of low-dimensional complex commutative algebras?
Let $\mathcal{A}$ be a finite-dimensional commutative associative unital $\mathbb{C}$-algebra. I am looking for a list (of further examples of) $\operatorname{Aut}_\mathbb{C}(\mathcal{A})$, the group ...
- 7,127
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0
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120
views
Multiplicativity of Euler–Poincaré characteristics of cohomology of pro-$p$ groups
While reading a paper, I found a mentioning that for an extension $1 \rightarrow H \rightarrow G \rightarrow N \rightarrow 1$ of pro-$p$ groups, the Euler–Poincaré characteristics $\chi(H)$, $\chi(G)$,...
- 1,013
1
vote
0
answers
203
views
Units in group rings in SAGE
Is there a recorded/known SAGE code to compute units in integral group rings for finite abelian groups ?
I would be happy with a code that only works for cyclic groups. I sort of know how to ...
- 15.9k
1
vote
0
answers
72
views
Bottleneck edge in lattice of subgroups
Let $G$ be a finite group. Define the bottleneck weight of a chain of subgroups $$\operatorname{id}=H_0 < H_1 < \ldots < H_n = G$$ to be the maximum value over the indices $[H_{i+1} : H_i]$ ...
- 396
1
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0
answers
97
views
A duality of finite groups coming from a surjective homomorphism with finite kernel of algebraic tori
$\newcommand{\Hom}{{\rm Hom}}
\newcommand{\Gm}{{{\mathbb G}_{m,{\Bbb C}}}}
\newcommand{\X}{{\sf X}}
$ I am looking for a reference for the following lemma (for which I know a proof):
Lemma.
Let $\...
- 14.2k