All Questions
Tagged with reference-request gr.group-theory
700 questions
10
votes
1
answer
382
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Wiener's axiomatization of the group law based on division
Gian-Carlo Rota wrote that [*]:
Wiener axiomatized the group law by taking $xy^{-1}$ as the basic operation, and his axiomatization is quite different from any of the other axiom systems for groups....
- 3,393
19
votes
2
answers
943
views
Reference for the triple covering of A_6
I would like to ask for a reference (book, paper ...) for the following nice construction, which I have found as an exercise in some notes of a course by R. Borcherds. For $n=6$ or $7$ (and only in ...
- 38k
1
vote
0
answers
62
views
Reference request for finite simple exceptional group of lie type $E_7(q)$ and its Schur covering group $2.E_7(q)$?
Does anyone have the paper named 'Génerateurs, relations et revêtements de groupes algébriques' written by Robert Steinberg in 1962, or any other reference for simple groups of Lie type $E_7(q)$ and ...
- 271
1
vote
1
answer
339
views
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
5
votes
2
answers
530
views
Wielandt automorphism tower theorem
I wanted to know if anyone can point me to an (ideally freely available) english translation of the proof of Wielandt's Automorphism Tower Theorem (1939).
The theorem states the following:
Given a ...
- 461
5
votes
1
answer
305
views
Schur covers of affine 2-transitive groups
I am interested in Schur covers of minimal 2-transitive groups. A theorem of Burnside gives that every finite 2-transitive group is either almost simple or affine. In the time since, these groups have ...
- 7,633
12
votes
1
answer
475
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Bi-orderability of Baumslag-Solitar group $\langle a,b \mid a^{-1} b^m a = b^n\rangle$ and of $\langle a,b \mid a^{-1} b a^m = b^n\rangle$
We say that a group $(A, \cdot)$ is bi-orderable if there exists a total order $\preceq$ on $A$ such that $xz \prec yz$ and $zx \prec zy$ for all $x,y,z \in A$ with $x \prec y$.
Let $m,n$ be non-zero ...
- 10.5k
18
votes
3
answers
745
views
Number of primitive $n$th roots with positive versus negative real parts
Does anyone know a reference to the following results, which I can prove, but I suspect may be known. Let $R(n)$ denote the number of primitive $n$th roots of unity with positive real part, and $L(n)$ ...
- 1,991
10
votes
1
answer
246
views
Naturally occurring, non-amenable Zappa-Szep products of discrete amenable groups?
We say $G$ is the Zappa-Szep product of two subgroups $K$ and $P$ if $K\cap P = \{e\}$ and the function $K\times P \to G$, $(k,p)\mapsto kp$, is bijective.
The Iwasawa decomposition shows that we can ...
- 25.8k
2
votes
0
answers
89
views
Name for a probability density ''symmetrized'' by a permutation group?
Let $p$ be a probability density function over random variable $X$, and $G$ a compact permutation group over the outcomes of $X$. For each $g\in G$, let $p_g$ indicate the probability density ...
- 695
19
votes
2
answers
1k
views
Does the amenability problem for Thompson's group $F$ predate 1980?
The first place where the amenability problem for Thompson's group $F$ appears in the literature is, I believe, 1980 in a problems article by Ross Geoghegan. I have heard, however, vague comments to ...
- 3,547
3
votes
0
answers
155
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A variant on the Higman-Thompson groups
Let $C = \mathbb{Z}/d\mathbb{Z}$ ($d \ge 0$).
Let $D = \langle a_c : c \in C, t \mid a^2_c = t^d = 1, ta_ct^{-1} = a_{c+1} \rangle$.
let $E$ be the subgroup generated by $\{a_c : c \in C\}$ and let $...
- 4,728
6
votes
1
answer
486
views
Fourth cohomology of the modular group
Is $H^4(PSL(2,\mathbb{Z}),\mathbb{Z})$ known? I ask this in response to the recent calculation of the same cohomology group for $\mathrm{Co}_0$ and $\mathrm{Co}_1$.
- 35.5k
12
votes
1
answer
1k
views
How can I have a copy of this old paper by Frobenius?
How can I have a copy of this old paper and a translation of it?
Frobenius, G. (1902). Uber primitive Gruppen des Grades n und der Klasse n - 1. S. B. Akad. Berlin 1902, 455-459.
- 229
7
votes
2
answers
301
views
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
10
votes
3
answers
855
views
History of profinite groups, when was it first mentioned? What was the original definition?
Searching left me hanging. One of my professors told me the definition using the topological properties was the first one but I cannot find any resources. Is that true? If not, how was it originally ...
- 373
3
votes
0
answers
58
views
Isoclinism for Lie groups: existing accounts of basic properties?
Philip Hall introduced the relation of isoclinism between two groups. One statement of the definition (not Hall's original statement) is to introduce a category whose objects are the canonical maps
$$...
- 25.8k
15
votes
0
answers
716
views
Is this "Homology" useful to study?
In the usual singular homology of a topological space $X$, one consider the free abelian group generated by all continuous maps from the standard simplex $\Delta^{n}$ to $X$.
Now we can ...
- 356
1
vote
1
answer
150
views
Lemma for proof of Jordan-Hölder Theorem [closed]
Lemma
Let $G$ be a group, $K \triangleleft G$ a normal subgroup and $H_j \triangleleft H_i$ two subgroups of $G$ with $H_j$ a normal subgroup of $H_i$.
Then there is an isomorphism
$$(H_i K)/(H_j K)...
- 708
6
votes
0
answers
164
views
Is a presentation of the hyperbolic orthogonal group of rank 2 over the integers known?
The hyperbolic orthogonal group $O_{g,g}(\mathbb{Z})$ often appears in the study of high-dimensional manifolds, see e.g. work of Kreck or Galatius and Randal-Williams. Let $H$ denote the lattice $\...
- 8,167
6
votes
3
answers
811
views
Enumerating cosets of the modular group
Suppose we chose the generators $f(z) = z+1, g(z) = z-1, h(z) = -1/z$ for the modular group $\Gamma$ (i.e. the group of fractional linear tranformations $z \mapsto (az +b)/(cz+d)$ with $a,b,c,d \in \...
- 4,304
1
vote
0
answers
76
views
Nomenclature: does this coset space have a name?
in my work I tripped on a specific coset space and before starting thinking about it by myself, I wanted to check the literature. However, I do not know if the object has a name (which makes ...
- 111
5
votes
1
answer
153
views
Group rings over central products
I have a proof of the following result but I was wondering if anyone had a reference for it. I have asked on math.stackexchange here but didn't receive any replies.
Let $G$ a finite group given by ...
- 79
11
votes
4
answers
2k
views
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
9
votes
3
answers
947
views
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
3
votes
0
answers
94
views
Clifford correspondence(s) from Fong-Reynolds theorem
The Fong-Reynolds theorem states a certain relationship between blocks of a normal subgroup $N\unlhd G$ and blocks of $G$, sometimes called the "Clifford correspondence for blocks". If one phrases ...
- 9,650
3
votes
1
answer
273
views
Preprint by Wall on Sjogren's theorem
In their account http://dx.doi.org/10.1016/0022-4049(87)90048-X of Sjogren's theorem, Cliff and Hartley refer to two articles:
[9] B. Hartley, A note on a lemma of Sjogren relating to. dimension ...
- 2,519
10
votes
4
answers
2k
views
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
2
votes
1
answer
190
views
Logic article on first-order invariants of abelian groups
I remember reading an article published in the 1970s by a Polish mathematician describing the first-order invariants of a torsion-free abelian group. I do not recollect the author's name, the title of ...
- 116
7
votes
2
answers
918
views
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
6
votes
1
answer
399
views
One-ended finitely presented subgroups of hyperbolic groups
In Hyperbolic groups (page 82), Gromov claims that, by a standard application of Thurston's method of geodesic (hyperbolic) simplices, it can be prove that a hyperbolic group contains finitely many ...
- 2,371
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
votes
3
answers
3k
views
Reference for Ring Structure on Group Cohomology
As a graded $\mathbb{Z}$-module, the structure of the group cohomology $H^{*}(\mathbb{Z}/n\mathbb{Z};\mathbb{Z})$ is extremely well-known. Yet, I am having difficulty finding a reference concerning ...
- 4,920
19
votes
2
answers
1k
views
Reference request for Plancherel measure
I need a good reference for the basic definitions of the dual of locally compact group (not necessarily abelian), its natural topology, $\sigma$-algebra, and the Plancherel measure on it (when they ...
- 26k
2
votes
1
answer
82
views
Structure of extensions arising in Lie approximation of connected groups
My imperfect understanding is that, by the work of various authors (Gleason, Yamabe, Montgomery, Zippin ...), the following result is known:
Let $G$ be a connected, locally compact, Hausdorff group, ...
- 25.8k
3
votes
4
answers
757
views
Lucido's three prime lemma
Let G be a finite solvable group. If p,q,r are distinct primes dividing |G|, then G contains an element of order the product of two of these three primes.
This is lucido's three prime lemma. I ...
4
votes
2
answers
704
views
Behaviour of cohomology groups under extension of scalars
Let $\hat{R}\to R$ be a homomorphism of commutative unital rings and let
$\hat{M}$ be an $\hat{R}G$-module for a group $G$. Does the $R$-module isomorphism $$H^n(G,\hat{M}\otimes R)\cong H^n(G,\hat{M}...
- 51
16
votes
2
answers
992
views
Maximal number of maximal subgroups
Let $G$ be a finite group. I want to find an upper bound on the number of the maximal subgroups. My questions is does it possible to prove that the number of maximal subgroups of any finite group $G$ ...
- 2,179
7
votes
2
answers
830
views
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
17
votes
3
answers
815
views
Does this subgroup of "even braids" have a name?
The full braid group on $n$ strands $B_n$ admits a surjective homomorphism $p\colon\thinspace B_n\to \Sigma_n$ onto the symmetric group on $n$ letters, which takes a braid to the induced permutation ...
- 35.9k
11
votes
0
answers
379
views
Amalgamated product of automatic groups
In Gersten's "Problems on Automatic Groups", Problem 14, he asks the following question: Let $G=A\ast_{C}B$ where $A$ and $B$ are automatic and $C$ is infinite cyclic. Is $G$ automatic?
Is this ...
- 525
11
votes
2
answers
778
views
History of Tarski's problems on free groups
As is known, Tarski posed his questions about first-order theories of non-abelian free groups around 1945. However, the questions were not published in his papers or books.
What is the original ...
- 893
3
votes
0
answers
136
views
Existence of loxodromic elements in certain subsets of $\text{PSL}_2(\mathbb C)$
Let $R$ be a subset of $\text{PSL}_2(\mathbb C)$ and consider its natural action on $\mathbb {CP}^1$. We say that $R$ is elementary if either $R$ is conjugated to a subset of $\text{SU(2)}$ or if ...
- 1,058
9
votes
3
answers
435
views
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
3
votes
1
answer
696
views
Unique factorization of finite groups under direct sum?
I am told that finite groups have unique factorization under direct product. That is, call a nontrivial group "indivisible" if it is not isomorphic to a direct product of nontrivial groups. Then ...
- 4,994
3
votes
0
answers
257
views
Braids with an infinite number of strings
Has anyone developed a theory for braids with an infinite number of strings?
- 1,181
21
votes
1
answer
564
views
Partitions of ${\rm Sym}(\mathbb{N})$ induced by convergent, but not absolutely convergent series
Let $(a_n) \subset \mathbb{R}$ be a sequence such that the series
$\sum_{n=1}^\infty a_n$ converges, but does not converge absolutely.
Then there is a partition of the symmetric group ${\rm Sym}(\...
- 19.6k
4
votes
0
answers
113
views
Liftings and splittings (reference request)
I'm writing a paper and, at a certain point, I need the following, rather elementary
Lemma. Assume that we have a commutative diagram of short exact sequences of groups of the form
Then the ...
- 66.3k
8
votes
2
answers
272
views
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
7
votes
1
answer
476
views
Centralizer of longest element in a finite irreducible Weyl group: related to folding of ADE graphs?
Say $(W,S)$ is a finite Coxeter group, such as a Weyl group (which satisfies an additional crystallographic condition). Assume also that $W$ is irreducible. Then it has a longest element $w_o$ ...
- 52.9k