All Questions
Tagged with reference-request gr.group-theory
700 questions
8
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2
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If a semigroup embeds into a group, then is it a subdirect product of groups?
The title has it all:
Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups?
If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
- 10.5k
8
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1
answer
322
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Does every cancellative duo semigroup embed into a group?
Prompted by the comments to a recent answer by YCor to a related question (here), I'd like to ask the following:
Q. Does every cancellative duo semigroup embed into a group?
A (multiplicatively ...
- 10.5k
8
votes
1
answer
227
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Non-finitely presented FP groups with cohomological dimension $2$
In this recent preprint, the authors construct a certain uncountable family of non-finitely presented FP groups. Recall that group is an FP group if the trivial $\mathbb Z[G]$-module $\mathbb Z$ has a ...
- 15.8k
8
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1
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319
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"Novelty" maximal subgroups in $S_n$
What are the maximal subgroups $M < S_n$ such that $M \cap A_n$ is not maximal in $A_n$?
Maximal subgroups of $S_n$ are described by the O'Nan-Scott theorem and very extensively studied in many ...
- 2,821
8
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1
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633
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Why is this group called "The Holomorph of a group"
Many years ago I found in google the notation "Holomorph of group". It is the semi direct product of $G$ with $Aut(G)$. Why is the term "Holomorph" used here, while it is usually used for complex ...
- 356
8
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1
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200
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For which planar topological spaces $Z$ does there exist a hyperbolic group $\Gamma$ with $\partial \Gamma \cong Z$?
Recall a topological space is called planar if it can be embedded in $S^2$. I'm interested in understanding hyperbolic groups with planar boundaries.
In [1], it is shown that if a one-ended hyperbolic ...
- 639
8
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1
answer
739
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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
8
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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
8
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2
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272
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Roller's problem on median groups
At the end of his dissertation Poc Sets, Median Algebras and Group Actions, Martin Roller asks
A group $G$ is called median if it acts freely and transitively on a median algebra. This is ...
- 2,371
8
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2
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840
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Intersection of conjugates of subgroups in free groups
I am looking for a reference for the following
Fact 1: if $A$ and $B$ are finitely generated subgroups of infinite index in a finitely generated free group $F$ then there exists $f \in F$ such that $...
- 3,181
8
votes
2
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3k
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Centralizers in GL(n,p)
There appear to be a number of rational canonical forms. The best thing about standards is how many there are to choose from. However, the standard I choose seems to have a centralizer that is ...
- 10.7k
8
votes
2
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482
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Parabolics and simple roots for a special unitary group: reference request
I am looking for a reference where the relative root system, the relative system of simple roots, and parabolic $\Bbb R$-subgroups for the real algebraic group ${\rm SU}(p,q)$ are explicitly computed.
...
- 14.2k
8
votes
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
8
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1
answer
513
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Equivariant (co)homology of flag manifolds, convolution algebra and nil hecke algebra?
For a complex reductive group $G$ and its Borel subgroup $B$, it seems to be well-known that the equivariant homology group $H^G_*(G/B\times G/B)$ forms a nil-Heck algebra
$$NH=\Bbbk[y_i,\partial_{j}]...
- 719
8
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1
answer
229
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Embedding abelian cancellative Hausdorff topological semigroups into abelian Hausdorff topological groups
An abelian cancellative semigroup embeds (via a semigroup monomorphism) into an abelian group. What about an abelian cancellative Hausdorff topological semigroup that does not embed (via a ...
- 10.5k
8
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0
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137
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Group presentations where discarding generators always yields a subgroup
Consider a group presentation $ \left< G= \left\lbrace \text{generators}\right\rbrace \, \middle|\, R = \left\lbrace \text{relators}\right\rbrace \right>$ (no finiteness assumptions). Given $S\...
- 1,005
8
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0
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125
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The conjugacy problem for two-relator groups
Is the conjugacy problem for two-relator groups known to be undecidable?
The word problem for two-relator groups is a famous open problem (appearing e.g. as Question 9.29 in the Kourovka notebook), ...
8
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0
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346
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A generalization of Feit–Thompson conjecture, for square-free integers
I asked the following question with my account that I have for these sites Mathematics Stack Exchange and MathOverflow. The bounty that I offered in MSE expired without answers. The post that I refer ...
8
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0
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545
views
What are the character tables of the finite unitary groups?
I need to know the (complex) character table of the finite unitary group $U_n(q)$. Lusztig and Srinivasan (1977) provide an abstract description, but parsing it requires a stronger background in ...
- 7,633
8
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0
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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
7
votes
3
answers
582
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Asymptotics for the number of abelian groups of order at most $x.$
The number of abelian groups of order $n$ (call it $a(n)$ is a studied subject (see http://oeis.org/A000688), but I can't seem to find any asymptotic results. Obviously, there is no asymptotic for $a(...
- 96.4k
7
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4
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1k
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Consequences of the Inverse Galois Problem
Are there any papers written about the consequences of the Inverse Galois Problem in case it is proved to be true or false?
We know a lot of things that would be true if the Riemann Hypothesis holds. ...
7
votes
3
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577
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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 ...
7
votes
2
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780
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Finite groups with a character having very few nonzero values?
A number theorist I know (who studies Galois representations) raised a question recently:
Which finite groups can have an irreducible character of degree at least 2 having only $n=2, 3$, or 4 ...
- 52.9k
7
votes
2
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529
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Telling group algebras apart
It's a big, famous, hard problem in operator algebras to determine if the von Neumann algebras $L(F_2)$ and $L(F_3)$ are isomorphic, or not. Here $F_n$ is the free group on n generators and $L(F_n)$ ...
- 18.7k
7
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2
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713
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Total sum of squares of characters of the symmetric group $\mathfrak{S}_n$
In my earlier MO post, I proposed the double sum $\sum_{\mu\vdash n}\sum_{\lambda\vdash n}\chi_{\mu}^{\lambda}$ regarding characters of the symmetric group $\mathfrak{S}_n$. Soon after, I started ...
- 43.2k
7
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1
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839
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Incomplete Failures of the Inverse Galois Problem
I thought of this question the other day and have not been able to get any traction on references or results along its lines, so I finally caved and decided to ask it here. I am no expert on Galois ...
- 5,191
7
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808
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Reference: Finite $p$-Groups
Hall and Blackburn made important contributions in the study of regular $p$-groups and $p$-groups of maximal class. From their work, one can understand that in the classification of groups of order $p^...
- 71
7
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3
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2k
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Bass-Serre theory textbook
I am a PhD freshman working on topological graph theory and geometric group theory. I would like to learn some Bass-Serre theory. What do you think is the best introductory textbook in this topic? ...
- 422
7
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2
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669
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Élie Cartan's paper "Les groupes réels simples, finis et continus" of 1914
Question 1.
Does Élie Cartan's paper
Les groupes réels simples, finis et continus,
Ann. Sci. École Norm. Sup. (3) 31 (1914), 263–355
contain a classification of $\Bbb C$-linear involutions of simple ...
- 14.2k
7
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2
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830
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Every free abelian group is slender, why?
Wikipedia states that every free abelian group is slender. Where can I find a proof?
If this is not trivial, then I will also need a reference to use in my paper.
- 504
7
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1
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697
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Growth of Thompson's group $F$
EDIT(August 2013): I accepted Mark's answer as being the state of art- there are two relevant references, one in the answer and one in the comments. The minimal growth rate of $F$ remains unknown with ...
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7
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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
7
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2
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417
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Catalogue of groups with short finite presentations
For various types of groups, there exist catalogues of those groups of the
particular type which are "small" in a certain sense. — For example:
The GAP Small Groups Library catalogizes ...
- 19.6k
7
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1
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281
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Question concerning the coefficients of block idempotents
Let $G$ be a finite group. Let $p$ be a prime number such that $p \mid |G|$.
Let Irr$(G)$ denote the set of ordinary irreducible characters of $G$.
For $\chi\in$ Irr$(G)$ define $e_{\chi} := \frac{\...
- 1,755
7
votes
2
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704
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Reference for nonlinearity of covers of $\operatorname{SL}(2,\mathbb R)$
It is known that no nontrivial connected cover of $\operatorname{SL}(2,\mathbb R)$ admits a faithful finite dimensional linear representation (see, for example, page 143 in Fulton-Harris and Exercise ...
- 1,782
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2
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917
views
Is this exact sequence known?
$\newcommand{\Tors}{{\rm Tors}}
\newcommand{\tf}{{\rm\, t.f.}}
\newcommand{\Gt}{{\Gamma\!,\,\Tors}}
\newcommand{\Gtf}{{\Gamma\!,\tf}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\Z}{{\mathbb Z}}
\...
- 14.2k
7
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1
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778
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Reference for the Brauer-Nesbitt theorem
In Wikipedia's article on the Brauer-Nesbitt theorem, they state that given a group $G$ and a field $E$, two semisimple representations $\rho_1,\rho_2 : G\longrightarrow \operatorname{GL}_n(E)$ are ...
- 7,527
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2
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415
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Graph which do not satisfy a pseudo-Poincaré inequality
Say that an infinite (connected) graph (with vertices of bounded degree) satisfies a $\ell_1$-pseudo-Poincaré inequality if there is a constant $C>0$ so that for any $n \in \mathbb{N}$ for any ...
- 4,432
7
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1
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313
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Subgroup ranks of the symmetric group
It's well known that every subgroup $G$ of $S_n$ has a generating set of size at most $n-1$ and that this generating set can be found algorithmically (by Jerrum's filter)
I have heard many times a ...
- 539
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2
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918
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Historical reference request on Nilpotent groups
From Wikipedia:
"Abelian groups were named after Norwegian mathematician Niels Henrik Abel by Camille Jordan because Abel found that the commutativity of the group of a polynomial implies that the ...
- 1,555
7
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1
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818
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Uncertainty principle for non-commutative groups
Is it true that for every group $G$ and $f\in \mathbb C[G]$ it holds that $$\dim(\mathbb C[G]*f)\mathop{supp}(f)\geq |G| ?$$
Here, $\mathbb C[G]$ is the group algebra, and by $\mathbb C[G]*f$ I ...
- 2,179
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1
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548
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The probability that two elements of a finite nonabelian simple group commute
It is mentioned in here (last paragraph of the first page) that Dixon proved the following result: the probability that two elements of a finite nonabelian simple group commute is at most $\frac{1}{12}...
- 71
7
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1
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616
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Looking for a modern source about Ulm Invariants
I'm looking for a modern, approachable text (preferably a website, textbook, or expository article, and preferably one easily available online or at a library) which can explain the concept of Ulm ...
- 1,979
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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
7
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1
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393
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Status of the Isomorphism problem for automatic groups?
I only ask because I don't know how to look for the answer.
- 1,987
7
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1
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958
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Groups whose normal subgroups form a chain with respect to inclusion
Let G be a finite group. In general, given two normal subgroups N and K of G, we need not have N < K or K < N. The easiest example is the Klein 4-group V4 and its subgroups of order 2. So assume ...
- 307
7
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1
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143
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Is there a pseudofinite group with a quantifier-free instance of the order property?
Recall that a group $G$ is pseudofinite if every first-order sentence $\varphi$ (in the language of groups) satisfied in $G$ is also satisfied in some finite group. Also recall that an instance of the ...
- 12.5k
7
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1
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169
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What is the maximal possible rank of a subgroup of a special linear group mod a prime?
Let $p$ be a prime number, and let $\mathbb{F}_p$ be the unique field of cardinality $p$.
What is $\max \{d(H) : H \leq \mathrm{SL}_3(\mathbb{F}_p)\}$?
Here we denote by $d(G)$ the smallest ...
- 11.3k
7
votes
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}(...
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