All Questions
Tagged with reference-request gr.group-theory
700 questions
3
votes
1
answer
309
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Intersection of maximal subgroups of PSL(2,q)
Let $G := PSL(2,2^r)$, and let $M$ be a maximal subgroup of $G$ isomorphic to $PSL(2,2^s)$. I need to compute $H := M \cap M^g$ for $g \in G-M$. It seems to me that $|H|$ must be $2^r, 2^r\pm 1$ or $...
- 307
1
vote
0
answers
150
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Reference request for the list of nilpotent subgroups of SU(2)?
It's not hard to show that all non-abelian nilpotent subgroups of $SU(2)$ are actually finite and in fact are conjugate to one of the generalized quaternion groups of order a power of two, $$Q_{2^n} =...
- 5,309
10
votes
2
answers
418
views
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
9
votes
1
answer
384
views
Reference for the fact that $SL_n(O_K)$ surjects onto $SL_n(O_K/I)$ for any ideal I
Let $\mathcal{O}_K$ be the ring of integers in an algebraic number field $K$ and let $I \subset \mathcal{O}_K$ be a nonzero proper ideal. It is not hard to see that the map $\text{SL}_n(\mathcal{O}_K)...
- 91
7
votes
1
answer
958
views
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
4
votes
1
answer
869
views
Finite Unipotent Groups: References
It will be a great pleasure for me if one can suggest "Survey Articles" on following topics related to the finite unipotent group $U(n,\mathbb{F}_q)$. (Thanks in advance!!!)
The number of ...
- 1,169
3
votes
2
answers
337
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Does every embedding of one unipotent group (over R) in another extend to an embedding of the respective upper triangular matrix groups?
Let $T^*$ denote upper triangular matrices (of the appropriate size) with positive diagonal entries and $\mathrm{UT}$ upper triangular matrices with all diagonal entries equal to 1.
Does every (...
- 1,089
1
vote
1
answer
804
views
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
6
votes
1
answer
236
views
Class number of Burnside groups
Let $B(m,n)$ be the Burnside group on $m$ generators of exponent $n$. Suppose the class number - the number of conjugacy classes - of $B(m,n)$ is finite. Does it imply that $B(m,n)$ is finite?
- 2,118
6
votes
1
answer
1k
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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
4
votes
1
answer
1k
views
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
5
votes
1
answer
413
views
Index of congruence modular subgroup of level (1,d)
Let $D = \text{diag}(1,d)\in M_{2}(\mathbb{Z})$ be a $2\times 2$ matrix, where $d$ is an odd integer. We define the subgroup $\Gamma_D\subset M_{4}(\mathbb{Z})$ as:
$$\Gamma_D := \left\lbrace R\in M_{...
- 8,998
7
votes
2
answers
571
views
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
13
votes
1
answer
1k
views
Convenient reference for subgroups of a finite semidirect product?
Given a finite group $G= H \ltimes N$ (with no particular constraints on $H, N$), it's probably been known for a long time how to describe efficiently the possible subgroups of $G$. A graduate ...
- 52.9k
2
votes
1
answer
387
views
Generators of the colored braid group (two colors), reference
I consider the group $B_{n,n}$, the braids, colored in two colors, say all odd strings are black and all even strings are white.
It is easy to find a set of generators for $B_{n,n}$:
$$
\begin{cases}
\...
- 5,063
0
votes
0
answers
105
views
specific qi on free groups
Let $F_n$ be the free group on $n$ generators, $n>1$.
If $\phi$ is a quasi-isometry (or a bijective bilipschitz equivalence) on $F_n$, then what can we say about the explicit form of $\phi$?
In ...
- 1,528
7
votes
0
answers
407
views
Linear vs smooth actions of finite groups on spheres, euclidean spaces and closed disks
I would like to know examples (with references, if possible) of the following:
(1) a finite group $G$ acting effectively and smoothly on a sphere $S^n$ (any $n$) but admitting no effective linear ...
12
votes
2
answers
771
views
Is "subamenable" the same as amenable?
Let $G$ be a finitely generated group. Does the following condition imply the amenability of $G$:
there is a function $\mu:\mathcal{P}(G) \to [0,1]$ such that:
(subadditive) $\mu(G) = 1$, $\mu(A \...
- 3,547
3
votes
2
answers
716
views
reference request for character theory of p-extraspecial groups
In a recent preprint 1302.1929 my co-authors and I make use of some results about the character theory of extraspecial groups. These were largely gleaned from haphazard Googling, comments made by ...
- 25.8k
4
votes
2
answers
1k
views
Reference request for two-generator subgroups of a free group
According to B. Fine, G. Rosenberger, On restricted Gromov groups, Comm. Algebra 20 (1992) 2171--2181, Gromov proved the following in his long article introducing word-hyperbolic groups:
Let $x$ ...
- 25.8k
3
votes
1
answer
1k
views
Finite subgroups of SO(3)
There are several proofs of the famous classification of finite subgroups of $SO(3)$. I heard that there is a "purely algebraic" one attributed to Camille Jordan. Does anybody know of a reference?
...
- 607
2
votes
1
answer
153
views
Collections in direct products and freeness
I am looking for references about the following type of questions:
Let $G$ and $H$ be two groups,
let $(g_i:i\in I)\subset G$ and $(h_i:i\in I)\subset H$ be collections of group elements,
and ...
user34424
1
vote
3
answers
840
views
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
12
votes
0
answers
424
views
Reference for class of involutions containing longest element of finite Coxeter group?
Consider a finite (say irreducible) Coxeter group $W$ with a fixed generator set $S$ and rank $n$. This is the same thing as a finite real reflection group, generated by a set of “simple” ...
- 52.9k
2
votes
0
answers
68
views
Centralizer/Normalizer of global sections of vector bundles on curves
Let $X$ be a smooth, projective curve of genus at least $2$ over $\mathbb{C}$ and $E$ be a vector bundle on $X$ of rank at least $2$. Given any point $x \in X$, denote by $S_x$ the image of the ...
- 1,981
7
votes
0
answers
252
views
A conjecture of Lubotzky on ranks of subgroups of special linear groups over the integers
In a 1985 paper named "Dimension function for discrete groups" Lubotzky conjectured that:
For any integer $n \geq 3$ the group $\mathrm{SL}_n(\mathbb{Z})$
contains infinitely many finite index ...
- 11.3k
2
votes
3
answers
2k
views
Infinite Field Theory and Category Theory
I should start by saying that I have not studied field theory in depth, so if this question is totally off base, I apologize. Something I noticed as I studied group theory is many concepts that were ...
- 5,759
9
votes
1
answer
1k
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Ping Pong and Free Group Factors
This question concerns alternative characterizations of free group factors. The ping pong lemma is a well-known criteria for the freeness of a group. I've often wondered if there is a ping pong like ...
- 7,067
2
votes
3
answers
530
views
Conjugacy classes in PSL(3,q) and PSU(3,q)
What are the conjugacy classes of $PSL (3,q)$ and $PSU(3,q)$?
- 31
8
votes
1
answer
739
views
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
0
votes
1
answer
363
views
Examples of groups such that order isomorphism of the subgroups of $G\times G$ and $H\times H$ does not imply isomorphism of $G$ and $H$
Let $G$ and $H$ be groups, $\operatorname{Sub}(G\times G)$ be the set of all subgroups of $G\times G$ and $\operatorname{Sub}(H\times H)$ be the set of all subgroups of $H\times H$. Assume there ...
- 1,145
2
votes
1
answer
310
views
Reference request for generalization of groups with out identity element?
In other words what do we call a magma which is associative and has divisibility property but not existence of identity? Or a groupoid when it loses the identity property?
A reference on such ...
- 143
5
votes
0
answers
96
views
Is every space group the symmetry group of some triply periodic minimal surface?
I know that there are a lot of TPMS with different symmetry groups. It seems like every space group is the symmetry group of some TPMS. But I can not find a reference that confirms this for all the ...
- 2,581
2
votes
3
answers
912
views
Reference on generators of 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
2
votes
1
answer
159
views
Counting elements with certain word length in abelian groups
Given a (finite) abelian group $G = \langle S \mid R \rangle$, has the problem of counting the number of elements which can be expressed as a word (in $S$) of length $\leq k$ been studied? If so, ...
- 121
6
votes
1
answer
828
views
Which functions are linear combinations of irreducible characters for a given field $\Bbbk$?
Let $G$ be a finite group. Then it is well known that a function $f\colon G\to \mathbb C$ is a linear combination of irreducible characters iff it is constant on conjugacy classes. What is the ...
- 38.6k
7
votes
2
answers
529
views
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
12
votes
1
answer
744
views
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
votes
1
answer
489
views
growth rate of $\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}$?
I am interested in the growth rate of this type of group: $G=\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}$, where $\sigma(a)=\begin{pmatrix}x&y\\z&w\end{pmatrix}\in SL_2(\mathbb{Z})$, where $a$ is ...
- 1,528
7
votes
1
answer
393
views
Status of the Isomorphism problem for automatic groups?
I only ask because I don't know how to look for the answer.
- 1,987
6
votes
1
answer
241
views
Is there an agreed-upon name for this type of subgroup?
In Embedding theorems for groups, (J. London Math. Soc. 34 1959 465–479.) Neumann and Neumann (NB: this is not the Higman-Neumann-Neumann paper of the same name) make the following definition.
...
- 335
8
votes
2
answers
3k
views
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
3
votes
0
answers
186
views
Which Dihedral Groups are $\text{CI}$-Groups?
Let $D_{n}$ denotes the dihedral group of order $2n$. Firstly, for self-referencing of the question, I give some definitions which are standard.
Let $G$ be a finite group. A subset $S$ of group $G$ ...
- 4,784
4
votes
1
answer
400
views
Speed of random walks in groups
I've seen some estimates for the decay in $d$ of the probability a SRW makes a distance $d$ in time $n$, but is there any reference for the "speed" of a random walk in a group? I'm interested mostly ...
- 4,432
1
vote
0
answers
96
views
Induced structure of topological group [closed]
If we consider a closed Jordan curve $\mathcal{C}$, I know that it's homeomorphic to the circle $S^1$. Now I take an homeomorphism $\phi:S^1\longrightarrow\mathcal{C}$ and this homeomorphism induces a ...
- 1,252
4
votes
1
answer
385
views
Divergence of geodesics in mapping class groups
I'm trying to learn some stuff about divergence of geodesics. Let $\gamma$ be a geodesic in a metric space $X$. The divergence of $\gamma$ is a function $f(r)$ for $r \ge 0$ such that $f(r)$ is the ...
- 619
3
votes
0
answers
282
views
Galois correspondence subgroups/subsystems
In this paper (1998) by M. Izumi, R. Longo, S. Popa, there is the following result (page 49) on compact groups:
Lemma 3.16. Let $G$ be a compact group and $Rep(G)$ the category of finite ...
2
votes
0
answers
190
views
Expository papers for Feit–Thompson Theorem [duplicate]
Feit–Thompson theorem states that every finite group of odd order is solvable.
Its proof is near 300 pages so it is definitely not an easy paper to read without a prior knowledge about the general ...
user82740
6
votes
1
answer
144
views
Continuity of conjugation actions of Polish groups
Let $G$ and $H$ be Polish groups and let $\psi: G \rightarrow H$ be a continuous injective homomorphism such that $\psi(G)$ is normal in $H$. Then $H$ acts on $G$ by conjugation via $\psi$, in other ...
- 4,728
4
votes
2
answers
998
views
A construction of generators of discrete subgroups of SL(2,R)
I know about geometrical method of construction of discrete subgroups of $SL(2,\mathbb{R})$ using Lobachevsky plane (e.g. B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, Modern Geometry --- Methods and ...
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