All Questions
Tagged with reference-request gr.group-theory
700 questions
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Generalizing groups via the Hall-Witt identity
In studying the integrability problem for Lie algebra representations, I have been led to wonder whether generalizing the notion of group by dropping associativity, while keeping the Hall-Witt ...
- 393
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4
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Textbook source for finite group properties deducible from character table?
Various questions have been posted on MO (some answered, some not) involving the character table of a finite group $G$ over a splitting field such as $\mathbb{C}$ of characteristic 0. My basic ...
- 52.9k
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votes
1
answer
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Automorphism group of factor groups
Let $G$ be a group and let $H$ be a factor group of $G$. Is there any result that relates $\operatorname{Aut}(G)$ (the automorphism group of $G$) and $\operatorname{Aut}(H)$?
As a very special case ...
- 1,108
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3
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Subgroups of GL_2 over a finite field
I've come across the phrase "by the classification of subgroups of $GL_2(F_q)$" in multiple papers, but never with a reference. Here $F_q$ is a finite field of size $q$. Does anyone know a good ...
- 101
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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
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2
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Algorithm for Brauer lifting via Brauer tree?
Background: Given a finite group $G$ and a prime $p$ dividing its order, Brauer theory compares the ordinary characters of $G$ with the Brauer characters arising from $p$-modular representations. On ...
- 52.9k
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Strong Atiyah conjecture
Who introduced the Strong Atiyah Conjecture?
Recall that the conjecture says the following. Let $G$ be a group, $A$ a $n\times n$-matrix over ${\mathbb Z}G$. We view $A$ as a bounded operator $l^2(...
user6976
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Proofs of the Stallings-Swan theorem
It is a well-known and deep${}^\ast$ theorem that if a group $G$ has cohomological dimension one then it must be free. This was proved in the late 60's by Stallings (for finitely generated groups) and ...
- 35.9k
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Transgressions commute with the Steenrod operations on the base and fiber in a central group extension?
The following sentence is quoted from the paper ON THE COHOMOLOGY OF SPLIT EXTENSIONS by D. J. BENSON AND M. FESHBACH:
In general, the differentials in the Lyndon-Hochschild-Serre spectral sequence
...
- 83
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645
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Reference request for the number of Sylow p-subgroups
Let $G$ be simple group of Lie type or Alternating group. I need reference for find the number of Sylow $p$-subgroup $G$ for every $p$. Thanks a lot.
- 221
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Connection between Deligne-Mostow monodromy and Gassner representation at roots of unity of the pure braid group
I am looking for a specific reference to the connection between [1] the Deligne-Mostow monodromy and [2] Gassner representation at roots of unity of the pure braid group. I have seen many references ...
- 11.2k
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2
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870
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A space of ideals
Definition: Let $R$ be a commutative ring with 1. Endow the power set $2^R$ with the product topology. The ideal space $\mathcal{I}(R)$ is defined to be subset of $2^R$ consisting of ideals, ...
- 25k
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'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
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Finite subgroups of the unimodular group
This is related to this MO question (and others as well).
Hoping that this will not turn out to be too broad, I would like to know about the 'state of the art' of:
1) The problem of classifying ...
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4
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Isomorphic but non-conjugate subgroups of $GL(n,\mathbb{Z})$ ?
I've been asked some questions by a friend interested in crystallography, and the following questions (I'm not an expert) came spontaneous to me:
1) Are there two finite subgroups $P,P'\subset\...
- 23.3k
5
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2
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281
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Doubly covering an even lattice
I have read that there is a way to construct a group which is a double cover of an even lattice. The very tantalizing thing about this is that if the even lattice is chosen to be the Leech lattice, ...
- 3,645
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2
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485
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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
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1
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589
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Commutator subgroups and normal $p$-complements
Let $G$ be a finite group with commutator subgroup $G'$. Let $p$ be a prime number.
Then $p \nmid |G'|$ if and only if $G$ has an abelian Sylow $p$-subgroup $P$ and normal $p$-complement $N$ (and in ...
- 1,661
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1
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Decomposition of semisimple Lie group into almost simple factors
Can anyone suggest a reference that defines or explains that a semisimple real Lie group can be decomposed into a product of almost simple factors? In some papers that I read recently people keep talk ...
- 511
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1
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540
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Cosets of groups of functions
Let's consider an interval $I\subseteq\mathbb R$, and let $\mathcal F(I)$ be the set of bijective functions $f:I\to I$ so that the graph of $f$ is a analytic curve in $I\times I$.
The set $\mathcal ...
- 4,312
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3
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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 ...
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Modular representations of the symplectic group
Let G=Sp(2m,2) be a finite symplectic group acting on $F_2^{2m}$. This group G acts 2-transitively on $\Omega_{+}$ and on $\Omega_{-}$. Let $F$ be an algebraic closure of $F_2$.
I am interested to ...
- 2,179
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303
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Automorphism group of algebraic function fields
Let $K$ be a finite field and let $F/K$ be a function field. Is it possible to deduce the genus of $F/K$ from the automorphism group of $G=Aut(F/K)$?
Is it possible to do so if we know that $|G|$ is ...
- 2,179
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1
answer
397
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Counting Nearest Neighbors that Stay Nearest Neighbors after Random Rearrangements
Imagine we are making necklaces with $n$ beads, each bead is a different color from all others. Let's say we make one necklace. If we make another necklace with the same $n$ differently colored ...
- 161
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2
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3k
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Examples of "Monster" groups
I am planning a talk for a general graduate student audience. The topic is exotic examples of countable discrete groups ("monsters"). Some examples of properties that I'm interested in are:
1.) Non-...
- 2,441
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2
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571
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abelian centralizers in almost simple groups
Hallo!
I'm looking for a reference. I'm sure that the information I need is already in the literature but I'm having some trouble to find it. Here is the question.
Let $S$ be a non-abelian finite ...
- 73
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1
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830
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Who proved that a group of polynomial growth has growth exactly polynomial?
I need to put a reference about the classical result that a f.g. group of polynomial growth has growth which is exactly polynomial.
Talking personally with people and also here in A question about ...
- 6,034
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1
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744
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Is the following construction of the 0-Hecke monoid (well) known?
Let W be a Coxeter group with Coxeter generators S. The corresponding 0-Hecke monoid H(W) has generating set S, the braid relations of W and the relations that each element of S is an idempotent. If ...
- 38.6k
6
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3
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505
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Irreducible mod-p representation of a semidirect product with trivial p-core
Consider the group $G=H\rtimes{}C$, where $H$ has order prime with $p$ and $C$ is cyclic of order $p^k$, and $C\rightarrow{}\mathrm{Aut}(H)$ is faithful (or equivalently $G$ has trivial $p$-core). ...
- 1,269
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F.p. groups where all elements of the same order are conjugate
The question I want to ask is related to the Boone-Higman conjecture (see
Embedding in f.p. simple groups for the details).
We discussed recently with Ievgen Bondarenko this conjecture and he ...
- 1,437
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2
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301
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Reference for projective covers of direct products of finite groups?
This concerns one of those "well known" facts, referred to in a recent preprint I've been looking at. In principle it's elementary, but I can't pin down an explicit textbook reference for it. ...
- 52.9k
3
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1
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608
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Amenability of Thompson's group looking at a 4-manifold having it as the fundamental group
Just for curiosity I have done a quick web-search and I have seen that some people are studying manifolds with amenable fundamental group. On the other hand, any finitely presented group and then, in ...
- 6,034
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Factorization of equivariant maps
Let $X$ be a finite set, $G$ a finite group and $M$ another Abelian
(multiplicative) group. Let us have a transitive (left) action $G
\times X \to X$ and an action $G \times M \to M$ by automorphisms.
...
- 3,110
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Unitary representations of the ax+b group: an accessible presentation
The "ax+b group" is the group of affine transformations of $\mathbb R$. It is a locally compact non unimodular group.
Its space of irreducible, continuous unitary representations has been described ...
- 9,486
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1
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446
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Radical of $F_p[SL(2,p)]$
Let $G=SL(2,p)$. Does anyone know what is the radical of the group algebra $F_p[G]$?
Does there exists any book/paper where it is calculated?
By radical here I mean maximal ideal I of $F_p[G]$ such ...
- 2,179
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4
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residually finite-by-$\mathbb{Z}$ groups are residually finite
I believe I read somewhere that residually finite-by-$\mathbb{Z}$ groups are residually finite. That is, if $N$ is residually finite with $G/N\cong \mathbb{Z}$ then $G$ is residually finite.
However, ...
- 2,821
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2
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Non-split groups
I am looking for a reference with definitions on what it means for an algebraic group to be split, quasi-split, and non-split. I would like to see some examples of the different "types".
Thanks,
Tom
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To what extent can one prescribe degrees of irreducible representations of a group?
Suppose one starts with an (infinite) multiset of positive integers $\mathcal{A} = \{a_i\}_{i\geq 0}$ such that:
$1=a_0\leq a_1\leq a_2\leq\ldots$
Can one always find a (necessarily infinite) group $...
- 5,191
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243
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Free Automorphisms
If $\varphi$ is an automorphism of $G = \langle x_1, \ldots, x_n; \mathbf{r}\rangle$ such that there exists an automorphism of $F(x_1, \ldots, x_n)$, $\overline{\varphi}$, with $$x_i\varphi=_G x_i\...
- 2,821
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The maximal order of an element in orthogonal groups over finite fields of characteristic 2
Let $q$ be a power of $2$ and let $(V,Q)$ be a
quadratic space of dimension $2m$ over $\mathbb{F}_q$. Up to isometry, we know that we have exactly two classes of such quadratic spaces: the plus type ...
- 7,609
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688
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Embedding in f.p. simple groups
Dear All!
At the time when Lyndon and Schupp wrote their book there was an open question:
Question: Does every finitely presented group with soluble word problem embed in a finitely presented simple ...
- 1,437
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645
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Popular level article on monster group
People who are not mathematicians (or high school students who are in maths) often become interested in what is the Monster Group - mainly because of unusual name. Since it's not my field, I'm able ...
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2
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418
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Do there exist groups with word problems in arbitrary P-degrees?
This has been posted on TCS stack exchange for a while here and hasn't gotten any answers, so I'm trying again here.
It has been known for a long time that, given any r.e. Turing degree, there is a ...
- 258
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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
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773
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Normal subgroups of projective special linear group over a ring
What are the normal subgroups of $PSL_2(\mathbb{Z}/p^n \mathbb{Z})$?
- 1,905
32
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0
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993
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Is there a Mathieu groupoid M_31?
I have read something which said that the large amount of common structure between the simple groups $SL(3,3)$ and $M_{11}$ indicated to Conway the possibility that the Mathieu groupoid $M_{13}$ might ...
- 3,645
3
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1
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149
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Reference for decomposition in invariants and derived subgroup in a semidirect product of abelian groups
Let $A$ and $B$ be finite abelian groups with coprime order, and let $G=A\rtimes{}B$ be a semidirect product, via any action. Let $C\subseteq{}A$ be the subgroup of the elements of $A$ which are fixed ...
- 1,269
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1
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499
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Posets of cosets and contractibility
For this question let $G$ be a group, perhaps infinite, and let $H_i$ for $i\in I$ be a (finite) family of subgroups closed under taking intersections. I am interested in the coset poset $\mathcal{C}(...
- 1,680
11
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2
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4k
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Orders of automorphism groups of p-groups
There is a theorem that says that if $p$ is a prime and $G$ is a $p$-group with $|G| = p^{n}$, $|Aut(G)|$ divides $\Pi_{k=0}^{n-1} (p^{n}-p^{k})$.
This theorem is sharp, since $\Pi_{k=0}^{n-1} (p^{n}-...
- 3,645
7
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3
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578
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Finitely presented groups which are not residually amenable
What are examples of finitely presented but not residually amenable groups?
Well, the examples that I want to have are simple f.p. groups as well as examples of non residually amenable groups arise ...