All Questions
Tagged with reference-request gr.group-theory
700 questions
5
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Dehn functions of Thompson's group $F$
It's well know that the first order Dehn function of $F$ is quadratic. Is a similar result known for its second-order, or even higher-order, Dehn function?
The second-order Dehn function of a group $...
- 51
5
votes
1
answer
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
4
votes
1
answer
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
4
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0
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176
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Is there a notion of "tame" representations of $GL_n(Z)$?
This is a followup to this question about the (left) noetherianity of the group ring of $GL_n(\mathbf{Z})$:
Does GL_n(Z) have a noetherian group ring?
Given that $\mathbf{Z}[GL_n(\mathbf{Z})]$ is ...
- 10.7k
0
votes
1
answer
305
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Embedding a semigroup into a divisible semigroup
The following is motivated by the fact that I'd like to have a way, much better if canonical, to isometrically embed a normed group into a normed divisible group. But semigroups are a much more ...
- 10.5k
2
votes
1
answer
253
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Generalising right-angled Artin groups
An Artin group $G$ is determined by its Coxeter matrix $M$. This is a symmetric $n \times n$ matrix with entries from $\lbrace 2, 3, \ldots, \infty \rbrace$ that determine the relations between the ...
- 3,165
2
votes
0
answers
132
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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
3
votes
1
answer
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
5
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57
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"Quasi-orthogonal" subgroup of a group with length?
For my project in bivariant K-theory for locally convex algebras, I'm looking how to call a particular notion of groups, too simple to be never considered elsewhere.
Let $G$ b a group with length $|\...
- 613
12
votes
1
answer
377
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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
3
votes
0
answers
156
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Cancellations in products of two elements of a hyperbolic group
Let $G$ be a non-abelian free group with the standard generating set and the corresponding word metric. If we take two elements $g,h\in G$ and compute their product $gh$, some letters might cancel, ...
- 637
0
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1
answer
227
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What is a "non-splitting covering" of a finite group?
Apologies if this is elementary, but I have never heard the terminology before:
What is a "non-splitting covering" of a finite group?
I encountered the term while reading this paper, in which ...
- 1,173
3
votes
1
answer
118
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On Groups of Maximal Class: Reference
I will be happy if one gives references (oncluding current research) for `classification' (structure) of $p$-groups of maximal class which contain abelian maximal subgroup (i.e. abelian subgroup of ...
- 1,169
2
votes
1
answer
163
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Amenable group rings embeddable in skew fields
I've made this question on math.stackexchange.com (also offering a bounty) but I did not receive any answer:
I'm looking for a reference of the following fact:
given a (countable?) amenable group $G$...
- 2,452
32
votes
0
answers
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
-1
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1
answer
214
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Question on the equal Sylow number in finite non-abelian simple group
let $G$ be a finite non-abelian simple group.If there exist $p$ and $q$ which are different prime numbers of $|G|$ such that $n_p(G)=n_q(G)$?
- 55
3
votes
0
answers
127
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"A locally dual polar space for the Monster"
I am currently looking at Ronan and Stroth's 1984 paper Minimal Parabolic Geometries for the Sporadic Groups. When considering the $3$-minimal parabolic system of $F_{1}$, they cite a preprint by ...
- 295
1
vote
1
answer
242
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Normal subgroups In a p-group [Reference?]
Dear Experts,
I'm a graduate student, dealing with group-theory.
In my current research, I used the bound "Alexander Gruber" wrote about in this post:
See Here
(Actually, I have just found out ...
- 35
1
vote
0
answers
66
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Quotient groups of "Abelian-times-compact", what are they called?
In what I am doing now this class of groups appears all the way: (Hausdorff) quotient groups of $A\times K$, where $A$ are locally compact abelian groups, and $K$ compact groups.
I wonder, if this ...
- 7,426
5
votes
0
answers
219
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Character tables of the p-core of the binary modular congruence group of p-power level
Let $p \geq 5$ be a prime and let $n$ be positive integer. In his Ph.D thesis (See The characters of binary modular congruence group, Bulletin of the
American Mathematical Society. 79 (1973), no. 4.), ...
- 2,396
3
votes
0
answers
264
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How do I determine the smallest dimension of an irreducible $\mathbb{F}_p[G]$-module with a prescribed trivial fixed point space?
This is a crosspost from MSE since I haven't found an answer there yet.
I am not very familiar with modular representation theory or Brauer theory yet, however lately I have needed to use $\mathbb{F}...
- 1,173
1
vote
0
answers
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 ...
3
votes
0
answers
209
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What is known about 2-modular representations of Ree groups of type $F_4$?
A too-vaguely worded question posted today about Suzuki and Ree groups reminds me to revisit a concern I never followed up years ago when assembling information about modular representations of finite ...
- 52.9k
-1
votes
1
answer
152
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Reference for the set of orders of its elements
I am looking for a reference for the maximal order of an element in PSL(2, $q$), where $q$ is prime power.
- 102
3
votes
0
answers
135
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Groups acting on non-locally-finite trees with independence and specified local actions
Suppose I have a biregular tree $T_{m, n}$ (not necessarily locally finite), with distinct cardinal numbers $m, n$, so Aut$(T_{m, n})$ acts on $T_{m, n}$ without inversion. Let $V_m$ be those vertices ...
- 313
6
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0
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276
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Walter Feit's program for characterizing $S_5$.
In Jacobson's Algebra Vol. I, there is a long, 10 part exercise which characterizes $S_5$ as isomorphic to any finite group having precisely two conjugacy classes, such that the centralizers of the ...
- 61
9
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329
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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
0
votes
0
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75
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The set of (property) elements of a locally compact group is closed
For which properties $(P)$ is the following statement known to be true?
In any locally compact group $G$, the elements of $G$ that satisfy $(P)$ form a closed subset of $G$. In other words, the ...
- 4,728
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
7
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0
answers
430
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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
0
votes
0
answers
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
1
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0
answers
252
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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
2
votes
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
0
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0
answers
289
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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
2
votes
1
answer
226
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Name of the Marshall Hall paper in which he proved that the intersection of all subgroups of a fixed finite index is again finite index?
can someone please tell me? I couldn't find a reference in the paper I was reading.
- 87
4
votes
0
answers
250
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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 ...
4
votes
1
answer
328
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Algebraic K-groups and braids
This is (I think) a reference request:
Are there calculations of any algebraic K-groups for the (group ring of) the Artin braid groups?
- 1,180
5
votes
1
answer
264
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Group not leaving subset invariant
Let $Y,X$ be two sets of size n,m. Let $Y\subset X$.
What is the maximal group(in size) $G< Sym(X)$ such that gY=Y imply that $g=1$?
Here I mean that the only permutation which permutes elements of ...
- 2,179
3
votes
0
answers
153
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On divisors occurring as subgroup sizes
Given a finite group $G$ define $D(G)$ to be the number of divisors $r$ of $|G|$ for which there exists a subgroup of $G$ of order $r$.
Clearly $D(G) \leq d(|G|)$, where $d(n)$ denotes the number of ...
- 731
3
votes
1
answer
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
3
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0
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144
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Infinitely generated powerful pro-$p$ groups
A pro-$p$ group of finite subgroup rank has an open subgroup $P$ that is uniformly powerful, meaning that $[P,P]$ is contained in the group generated by $2p$-th powers in $P$, and raising elements to ...
- 4,728
2
votes
1
answer
274
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virtual chain conditions in groups
In group theory, it's often very useful to know whether a family of subgroups (eg normal subgroups, Zariski-closed subgroups, ...) satisfies an ascending chain condition or a descending chain ...
- 4,728
4
votes
1
answer
371
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Normal subgroups of binary polyhedral groups (reference request)
The binary polyhedral groups are finite subgroups of the quaternions corresponding (via McKay's ADE classification) to the $E$ series of affine Dynkin diagrams. They are also the lifts to $\mathrm{...
- 33.3k
4
votes
0
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237
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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
2
votes
0
answers
41
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Central automorphisms of groups act transitively on Krull-Schmidt decompositions
(Cross posted from math.SE)
I'm looking for a modern reference to the subject line, preferably one that doesn't use Ore's generalizations to modular lattices.
To clarify terminology...
Suppose we ...
- 822
5
votes
0
answers
107
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A dynamical property of automorphisms of a locally compact group
Let $G$ be a Hausdorff locally compact group and let $\alpha$ be an automorphism of $G$. Say $\alpha$ is (forwards) topologically recurrent if for all $g \in G$ and all neighbourhoods $O$ of $g$, the ...
- 4,728
1
vote
0
answers
125
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Isomorphisms of group extensions arising from antisymmetric forms
Let $V,W$ be topological vector spaces and fix continuous antisymmetric bilinear forms $\omega_1:V\times V\to \mathbb{R}$, $\omega_2:W\times W\to\mathbb{R}$. Since $\omega_1$ is a 2-cocycle (in fact ...
- 1,411
2
votes
1
answer
185
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Kurosh radical theory for topological groups?
Does anyone know if there has been much work done on radical and semisimple classes in the sense of Kurosh within the category of topological groups (or subcategories thereof)? For instance, for a ...
- 4,728
8
votes
0
answers
252
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Amenability versus the ideal of wandering sets
Let $G$ be a finitely generated group acting on a set $S$ (on the right). Define the heirarchy of "marginal sets" as follows:
The emptyset is 0-marginal.
A set E is $(k+1)$-marginal if $E$ can be ...
- 3,547
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 ...
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