All Questions
Tagged with reference-request gr.group-theory
700 questions
8
votes
4
answers
659
views
Normal Covering of a Finite Group
Suppose $G$ is a finite group and $N_1, N_2, \cdots, N_k$ are proper normal subgroups of $G$. The set $\{ N_1, \cdots, N_k\}$ is called a normal cover for $G$, if $G = \cup_{i=1}^kN_i$. I need to the ...
1
vote
1
answer
115
views
Bounds for Khukhro-Makarenko theorems
Let’s define the set of outer-commutator group words $OC \subset F_\infty = F[x_0, x_1, …, x_n, …]$ using the following recurrence:
$$\forall i \in \mathbb{N} \text{ } x_i \in OC$$
$$\forall u, v \...
- 2,618
1
vote
1
answer
194
views
Reference request: The commensurator of an arithmetic lattice is a simple group
I am interested in a reference and proof for some version of the following (folklore?) statement:
``Let $G$ be a (semi)simple Lie group (with no compact factors and trivial centre) and let $\Gamma$ ...
- 855
10
votes
1
answer
269
views
Edge-transitive Cayley graphs of $S_n$
I came across the following question which I haven't seen before:
Question. Fix $k\ge 3$. For infinitely many $n$, does there exists a generating set $\langle R_n \rangle = S_n$, $|R_n|=k$, such ...
- 17k
9
votes
1
answer
893
views
Kaplansky conjecture (consequences)
The Kaplansky conjecture says that: for any field $F$ and any torsion free group $G$, the group ring $F[G]$ does not have nontrivial idempotent elements.
Questions
Do we assume that $F$ has any ...
- 641
17
votes
1
answer
2k
views
A synopsis of Adyan’s solution to the general Burnside problem?
Where can I find a high-level overview
of Adyan’s original proof of the existence of finitely generated infinite groups with finite exponent?
Additionally:
If possible, would an expert please ...
- 7,067
22
votes
2
answers
2k
views
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
4
votes
0
answers
322
views
Steinberg relations for elementary subgroup of a Chevalley group over an arbitrary ring
Given a semisimple Lie algebra $\frak{g}$ of type $\Phi$ with a Lie algebra representation $\rho:\frak{g}\to \frak{gl}(v)$ and an arbitrary commutative ring one can associate the following gadgets:
...
- 435
10
votes
3
answers
1k
views
Natural associative law for a ternary "group"?
Suppose one were to define a group-like structure based on a set $G$
with a ternary (rather than binary) operator $g( a, b, c ) = \left< a, b, c \right>$.
One possible definition for the ...
- 151k
24
votes
2
answers
3k
views
Does any textbook take this approach to the isomorphism theorems?
Below, I present an outline of a proof of the first isomorphism theorem for groups. This is how I usually think of the first isomorphism theorem for ______________, but groups will get the points ...
- 12.1k
2
votes
1
answer
847
views
Why Triangle of Mahonian numbers T(n,k) forms the rank of the vector space?
I am looking for an explanation of why Triangle of Mahonian numbers T(n,k) form the rank of the vector space $H^k(GL_n/B)$?
With respect to the property of Kendall-Mann numbers where the statement ...
30
votes
1
answer
592
views
Guess that group via product queries
Suppose someone (person B) knows a finite group $G$ of order $n$.
You (person A) know only the order $n$,
and that $1$ is the name of the identity element.
The group elements are named $1,2,\ldots,n$ ...
- 151k
1
vote
1
answer
312
views
Topological Invariants for Group
Let $\mathbf{Grp}$ be the category of groups and $\mathbf{Top}$ be the category of topological spaces. To each group $(G, \circ_G)$, we can associate a topological space $(G,\tau_G)$ the basis for ...
user57432
2
votes
1
answer
151
views
Topological analogue of an FC group?
By definition, a group is FC if all its conjugacy classes are finite.
Has anything been published about a generalization of the FC property for topological groups?
- 51
6
votes
1
answer
405
views
An algorithm determining whether two subgroups of a finitely generated free group are automorphic
In the book Lyndon, Schupp, Combinatorial Group Theory, P.30 in the edition from 2000 They mention an unpublished work by Waldhausen that is said to give an algorithm to determine whether two ...
- 193
12
votes
2
answers
3k
views
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
7
votes
1
answer
237
views
Finite subgroups of $PSU(3)$
I'm looking for a reference to a classification or description of finite subgroups of $SU(3)$ that contain the center, or equivalently $PSU(3)$. Can anyone point me in the right direction?
- 71
2
votes
0
answers
133
views
Looking for the multiplicity-free paper by N.Inglis
I'm looking for the paper "Multiplicity-free permutation characters, distance-transitive graphs and classical groups, PhD Thesis, University of Cambridge, 1986" by Nicholas Francis John Inglis. The ...
- 9,501
0
votes
0
answers
186
views
Subset of reals associated to pairs of matrices in $\mathrm{SL}(2,\mathbb{R})$
Let $\Gamma$ be a subgroup of $\mathrm{SL}(2,\mathbb{R})$. I would like to ask if there is any research on the following set:
$$\Gamma*\Gamma:=\bigg\{\dfrac{(a+b)(a'+b')}{(c+d)(c'+d')}\bigg|\begin{...
- 333
9
votes
1
answer
495
views
Divergence of Groups and Metric Spaces
Several papers, including this and this claim that divergence of finitely generated groups and metric spaces have been introduced by Misha Gromov in his paper "Asymptotic invariants of infinite groups"...
user6976
1
vote
0
answers
126
views
Algebraic structures on graphs
There are many algebraic structures linked to graphs.
For example one can find zero divisor graphs $[1]$, $[2]$ and many other graphs.
Does there exist any survey paper which characterizes all the ...
- 444
8
votes
0
answers
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
4
votes
0
answers
174
views
Algebraic varieties associated to finite groups
Have the following equations been studied in the literature?
Let $G$ be a finite group.
Then I am looking for functions $f : G \rightarrow \mathbb{C}~ \backslash \left\lbrace 0 \right\rbrace $ such ...
- 2,753
17
votes
1
answer
575
views
Group cochains invariant under the action of the symmetric group
Let $G$ be a finite group and $A$ an abelian group. Recall the cochain groups
$$
C^k = \{f: G^k \to A\}
$$
and the coboundary map
$$
\delta : C^k \to C^{k+1}
$$
$$
(\delta f)(g_1, \ldots, ...
- 12.8k
0
votes
0
answers
79
views
What do Sylow 2-subgroups look like for Schur covering groups of finite simple groups?
What do Sylow 2-subgroups look like for Schur covering groups of finite simple groups?
Are there any references in which we can find the stucture of Sylow 2-subgroups of Schur covering groups of ...
- 271
3
votes
0
answers
98
views
Hales' generalization of the stacked bases theorem (seeking a proof)
In his paper Analogues of the stacked bases theorem, published in the proceedings of a 1976 conference, A.W. Hales claimed some interesting generalizations of the stacked bases theorem for abelian ...
- 2,992
10
votes
3
answers
1k
views
Random walks and Lyapunov exponents
Given a sequence $Y_1, Y_2, \dots$ of i.i.d. matrices in $\mathrm{GL}_n(\mathbb R)$, there is a theorem of Furstenberg and Kesten which says that if $\mathbb E(\log\|Y_1\|)$ is finite, there exists a ...
- 315
30
votes
0
answers
999
views
Follow-up to Steinberg's problem (12) in his 1966 ICM talk?
Steinberg's lecture at the 1966 ICM in Moscow here surveyed his work on regular elements of semisimple algebraic groups, while also formulating a number of then-open questions as "problems" (...
- 52.9k
4
votes
1
answer
119
views
Reference for nonquasi-split groups of type $E_6$ and $E_7$ over local fields
The semisimple groups over a local field have been classified by Tits, cf. [1] "Classification of algebraic semisimple groups" in Boulder and [2] "Reductive groups over local fields" in Corvallis.
In ...
- 991
4
votes
3
answers
282
views
Conjugacy in right-angled Artin groups
I am looking for a reference containing the following result:
Let $a$ and $b$ be two elements of a right-angled Artin group $A$. Assume that $a$ and $b$ have minimal length (with respect to the ...
- 8,401
4
votes
2
answers
1k
views
Every abelian torsion-free group is strictly totally orderable (via the compactness theorem)
Let $\mathbb G = (G, +)$ be a group. We say that $\mathbb G$ is strictly totally orderable (others would say bi-orderable) if there exists a total order $\preceq$ on $G$ such that $x+z \prec y + z$ ...
- 10.5k
6
votes
4
answers
2k
views
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
2
votes
0
answers
137
views
Time complexity of randomized algorithm: right-multiplying by random elements $z_i$ from a group $H$ to achieve $H$-invariance
Note: This question was inspired by a related question about the Quantum Merlin Arthur (QMA) complexity class on Quantum Computing Stack Exchange. I was deliberating whether to ask this on CS Theory ...
- 269
2
votes
0
answers
135
views
English translation of Fouxe-Rabinovitch paper
Is there somewhere an english translation of Fouxe-Rabinovitch's papers
"D. I. Fouxe-Rabinovitch, Uber die Automorphismengruppen ¨
der freien Produkte. II, Rec. Math. [Mat. Sbornik] N.S., 1941,
...
- 257
4
votes
0
answers
124
views
Abelian-by-cyclic subgroups of exponential growth solvable groups
I am currently looking for a reference to a proof (or counterexample) to the following statement:
Statement: Assume $G$ is a finitely generated solvable group of exponential growth, then there is a ...
- 4,432
2
votes
0
answers
143
views
Regular epi- and mono-morphisms for locally compact (Hausdorff) groups
I am interested in what the regular monomorphisms are in the category of locally compact (for me, always Hausdorff) groups (with continuous group homomorphisms).
It is easy to see that the equaliser (...
- 18.7k
10
votes
1
answer
257
views
Low dimensional representations of $SL_n(\mathbb{Z}/p^\ell \mathbb{Z})$
When $\ell = 1$ I know that the smallest non-trivial irreducible complex representations of $SL_n(\mathbb{Z}/p\mathbb{Z})$ has dimension $\frac{p^n - 1}{p-1} - 1$ (with maybe some exceptions for ...
- 2,242
10
votes
2
answers
538
views
Is there an analogue of Mostow-Palais equivariant embedding theorem for noncompact groups
Let $M$ be a (Hausdorff) smooth compact manifold and $G$ a Lie group acting smoothly on $M$. If $G$ is compact then, by Mostow-Palais theorem, there exists an equivariant smooth embedding $M\to {\...
- 31.2k
2
votes
0
answers
115
views
The structure of $PSL_2$ over the p-adic integers
As is well known, the group $\mathrm{PSL}(2,\mathbf{Z})$ is isomorphic to the free product of two cyclic groups of orders 2 and 3.
Is there a similar description of the projective special linear ...
- 171
2
votes
0
answers
75
views
Reference request for certain kinds of FC-groups
This reference request has a general part and a specific part. In both cases, it feels like I could make faster progress in searching the literature or reconstructing arguments, if I knew the correct ...
- 25.8k
2
votes
1
answer
439
views
Quotient groups of the lower central series of a surface group
In the answer to MO question 132247, it is possible to find a nice computation of the quotient groups of the lower central series of a finitely generated free group.
Q. What are the quotient ...
- 66.3k
9
votes
2
answers
701
views
Reference for embeddings of reflection groups (related to folding ADE Coxeter graphs)?
There are a couple of indirect methods, using Lie theory or Springer's general theory of regular elements in (real, complex) reflection groups, to construct natural embeddings among certain Weyl ...
- 52.9k
7
votes
4
answers
1k
views
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. ...
3
votes
0
answers
165
views
First reference to the Tits alternative
As we know, the "Tits alternative" is a theorem relating to finitely generated linear groups.
I was curious as to where in the literature the Tits alternative is first referred to by this name, as I ...
- 5,146
3
votes
0
answers
199
views
Generalization of normal subgroup
I am wondering whether the following concept appears in the group theory literature under some (perhaps different) name. Let $G$ be a group and let $A,B$ be subgroups of $G$.
Definition. Say that $(...
- 1,329
15
votes
4
answers
1k
views
Realizable Order Sequences for Finite Groups
My post is motivated at least in part by this MO question.
Has there been any work done on realizable order sequences for finite groups? By an "order sequence" I mean a non-decreasing list of the ...
- 7,831
6
votes
4
answers
561
views
SO$(4)$ (& SO$(n)$) characterization?
I believe it is the case that
any finite subgroup of SO$(3)$
(the $3 \times 3$ orthogonal matrices of determinant $1$)
is either a cyclic group $C_n$,
or a dihedral group $D_n$, or one of the groups ...
- 151k
5
votes
1
answer
163
views
Characteristically simple locally compact abelian groups
Say a topological group $G$ is topologically characteristically simple if there does not exist a closed subgroup $1 < K < G$ such that $K$ is invariant under all automorphisms of $G$ (here `...
- 4,728
5
votes
2
answers
441
views
Reference Request: Derived group of $\mathscr R_u(B)$
Let $G$ be a connected, reductive group over an algebraically closed field $k$. Let $B$ be a Borel subgroup with maximal torus $T$ and unipotent radical $U$. Let $\Phi^+ = \Phi(B,T)$ and $\Delta$ ...
- 6,180
6
votes
1
answer
366
views
Group of order $5p^aq^b$
In Lectures by Dan Bump on Modular representation theory,
Theorem 13.14 states that whenever $G$ is a non-abelian simple group of order $|G|=p^aq^br$ for distinct primes $p$,$q$, and $r$, every $r$-...
- 367