All Questions
Tagged with reference-request gr.group-theory
700 questions
6
votes
0
answers
139
views
Equation in a nilpotent group
Let $G$ be a nilpotent group of class at most $r$
(that is, $\gamma^{r+1}G=1$).
Let elements $g_1,\dotsc,g_n\in G$ be fixed.
We are interested in the set $V\subseteq\mathbb Z^n$ of solutions $x=(x_1,\...
- 809
6
votes
0
answers
121
views
Sylow subgroups of the restricted Burnside group $\mathrm{RB}(d,n)$?
$\DeclareMathOperator\RB{RB}$What is known about the Sylow subgroups of the restricted Burnside groups $\RB(d,n)$ ?
I am looking for a reference.
In fact my question is slightly more general. Recall ...
- 61
6
votes
0
answers
236
views
Group homomorphism from $\mathrm{GL}_p$ to $\mathrm{SL}_p$ in characteristic $p$
If $k$ is a commutative field of characteristic $p>0$, then the map
$$ \theta \colon \mathrm{GL}_p(k) \to \mathrm{SL}_p(k) \colon A = (a_{ij}) \mapsto (\det A)^{-1} (a_{ij}^p) $$
is a group ...
- 6,614
6
votes
0
answers
245
views
A group action on another group action quotient: how to best describe the resulting structure and does it have a name?
Suppose I have an action $\alpha:G\times X\to X$ of a group $G$ on a set $X$ and, on top of that, an action $\beta:H\times(X/G)\to(X/G)$ of another group $H$ on the set of $G$-orbits.
Is there a nice ...
- 18.4k
6
votes
0
answers
234
views
Nascent formal group law
$\DeclareMathOperator\FGL{FGL}$The formal group law (cf. Wikipedia, Ex. 1.6 of nLab, Hazewinkel) derived from an analytic function or formal series $f(x) = x + a_2 x^2 + a_3 x^3 + ...$ and its formal ...
- 10.5k
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
0
answers
225
views
Parshin's buildings for higher local fields
What is the status of the theory of buildings for higher local fields?
I know that there are some papers of Parshin, in which he describes some examples, like $PGL_2$ and $PGL_3$ over two-...
- 17.4k
6
votes
0
answers
276
views
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
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
5
votes
5
answers
873
views
Green polynomials
Is there any software for calculating Green polynomials (of type A)? Or, at least, where can I find tables of Green polynomials? Also, I would be interested in some formulas for Green polynomials in ...
- 1,590
5
votes
3
answers
677
views
Spectrum and scheme of the commutative group-algebra of an abelian group.
The group-algebra of an abelian group is commutative, so we can consider the spectrum of this algebra. Are there any information about the abelian group that we can obtain from such considerations? ...
- 263
5
votes
2
answers
984
views
Automorphism Group of some Classical groups
Hi All,
I would like to know the Automorphism group of some simple classical groups, such as PSL(n,q) or some PSU or PSp groups. Could you please give me some recommended books or papers then? I ...
- 233
5
votes
3
answers
851
views
What are some examples of non-commutative $\mathbb{Q}$-monoids and/or $\mathbb{R}$-monoids?
Definition 0. Let $R$ denote a commutative semiring with $0$ and $1$. By an $R$-monoid, I mean a monoid $M$ equipped with an action $R \times M \rightarrow M$ denoted $r,m \mapsto m^r$, satisfying the ...
- 3,793
5
votes
3
answers
230
views
Maximal subgroups of odd index in $\mathrm{PSL}(3,q)$
Let $G = \mathrm{PSL}(3,q)$ for $q$ odd. I am trying to understand a question that involves understanding the subgroups that contain a Sylow $2$-subgroup, and in particular, are subgroups of odd index ...
- 366
5
votes
2
answers
564
views
Finite groups factorized into two simple alternating groups
My research is somehow related to the following question :
Describe and classify all finite groups $G$ such that $G=HK$ with $H \cap K=1$, where $H \cong A_m$ and $K \cong A_n$ for some integers $m, ...
- 490
5
votes
2
answers
452
views
"geometric" description of the algebra of central functions on a Lie group
I am looking for a a description of the algebra of continuous central functions on a group, say a compact simple Lie group $G$, as the algebra of all continuous functions on a "nice" compact Hausdorff ...
- 2,201
5
votes
2
answers
332
views
Embedding $G$ in a $Z(G)$ extension of $\operatorname{Aut}G$
This question follows up a question I asked on math.SE. This is a refinement and a reference request.
For what groups $G$ does there exist a $Z(G)$-extension of $\operatorname{Aut}G$ (call it $\tilde ...
- 2,851
5
votes
1
answer
329
views
A hyperbolic group with a small profinite completion
Is there a finitely generated non-elementary word hyperbolic group the profinite completion of which is known (or conjectured) to be rather restricted, that is: abelian, pro-$p$, virtually prosolvable,...
- 11.3k
5
votes
1
answer
458
views
Lower Central Series of Pure Braid Groups?
What is the lower central series $\Gamma_k(P_n)$, where $P_n$ is the pure braid group with $n$ strands? We know that $P_n$ is generated by elements $A_{i,j}$; do we know the generators of $\Gamma_k(...
- 1,108
5
votes
3
answers
645
views
Reference request for the number of Sylow p-subgroups
Let $G$ be simple group of Lie type or Alternating group. I need reference for find the number of Sylow $p$-subgroup $G$ for every $p$. Thanks a lot.
- 221
5
votes
2
answers
439
views
Condition(s) for the full autormophism group $\operatorname{Aut}(C(G, S))$ of the Cayley graph of $G$ to be isomorphic to $G$
If $\Gamma = C(G, S)$ is the (undirected) Cayley graph of a finite group $G$ with generating set $S$, then $G \le \operatorname{Aut}(\Gamma)$, the "full" automorphism group of $\Gamma$.
When is it ...
- 178
5
votes
2
answers
346
views
Reference request: A theorem by S. Garrison
A theorem by S. Garrison states that if $G$ is a finite solvable group and $|cd(G)| = 4$ then $dl(G)\leq |cd(G)|$ (the Taketa inequality, which is conjectured to hold for all finite solvable groups). ...
- 2,508
5
votes
1
answer
175
views
Growth of the word norm for elementary matrices in $\rm SL_3 (\mathbb{Z})$
This is a reference request, since the answer is probably well known, but I could not find it.
Given a finitely generated group $\Gamma$ with a generating set $S$, define the word norm $l = l_S : \...
- 1,163
5
votes
1
answer
152
views
How to find a finite splitting field $K$ for $G$ such that every indecomposable $KG$-module is absolutely indecomposable
Let $G$ be a finite group and let $k$ be a finite field with char$(k)=p$ such that $p\mid |G|$.
If $k$ is a splitting field for $G$, then, no matter which splitting field we take, after extending ...
- 1,755
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
884
views
solvable word problem without algorithm
Let $G$ be a finitely generated group. I wonder if there are examples where:
1) The word problem is known to be solvable in $G$ but there is no algorithm known.
2) The word problem is known to be ...
- 829
5
votes
1
answer
906
views
Boundaries of relatively hyperbolic groups
When the interior of an n-manifold $M$ has a pinched negative curvature metric of finite volume, then its fundamental group $\Gamma=\pi_1M$ is relatively hyperbolic relative to the parabolic groups $\...
- 10.4k
5
votes
1
answer
365
views
Number of $k$-tuples of elements generating a cyclic group
Let $k$, $m$ be natural numbers, and $C_m:=\mathbb{Z}/ m \mathbb{Z}$ be the cyclic group of order $m$.
Let $N_{k, \, m}$ be the cardinality of the following set: $$\{(a_1, \ldots, a_k) \in (C_m)^k \; ...
- 66.3k
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
5
votes
1
answer
769
views
F.p. groups where all elements of the same order are conjugate
The question I want to ask is related to the Boone-Higman conjecture (see
Embedding in f.p. simple groups for the details).
We discussed recently with Ievgen Bondarenko this conjecture and he ...
- 1,437
5
votes
1
answer
279
views
Permanent of a Kronecker product of matrices
It is well known that $\det(A \otimes B) = \det(A)^m \det(B)^n$ when $A$ and $B$ are square matrices of size $n$ and $m$ where $\otimes$ denotes the Kronecker product.
Question: Is there a similar ...
- 26.5k
5
votes
1
answer
390
views
mod p (odd) cohomology of dihedral groups
I've been trying to find the cohomology for the trivial module for $\operatorname{PSL}_2(r^n)$ over $\mathbb{F}_p$ for $2 \neq p \neq r$ and have managed to reduce this to the cohomology of a maximal ...
- 115
5
votes
1
answer
152
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
5
votes
1
answer
472
views
Countable reduced abelian group containing all countable reduced abelian groups
Is there a countable abelian group for which its subgroups are exactly all of the countable "reduced" abelian groups? (Reduced means that its divisible subgroup is zero)
Is the following group a ...
- 165
5
votes
2
answers
281
views
Doubly covering an even lattice
I have read that there is a way to construct a group which is a double cover of an even lattice. The very tantalizing thing about this is that if the even lattice is chosen to be the Leech lattice, ...
- 3,645
5
votes
1
answer
342
views
Product of all conjugacy classes
Related to this post of my coauthor Sebastien, I should also mention that one can also prove the following dual result:
For any finite group G, the following identity holds:
$$
\left(\prod_{j=0}^m \...
- 887
5
votes
1
answer
175
views
Looking for a BAMS text about the group with commutation relations defined using meaningful words
What I definitely remember is that I saw a description of the following in the Bulletin of the American Mathematical Society, sometime in eighties (or maybe nineties?)
One considers the group ...
- 18.4k
5
votes
1
answer
597
views
Can every cancellative invertible-free monoid be embedded in a group?
A monoid is invertible-free if $xy=1$ implies $x=y=1$ for all $x,y$.
Question: Can every cancellative invertible-free monoid be embedded in a group?
I'm fairly sure that a quotient of the free product ...
5
votes
2
answers
1k
views
Malcev's paper "On a class of homogeneous spaces" in English
I am struggling to find the English translation of Malcev's paper "On a class of homogenous spaces" providing foundational material for nil-manifolds. To be precise this paper: Malcev, A. I. On a ...
- 51
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
1
answer
889
views
A generalized Burnside's lemma
Let $G$ be a finite group acting on a set $X$, and let $S\subseteq G$ be a union of conjugacy classes. Then I believe I can prove:
$$ \sum_{[x]\in X/G} \frac{|G_x \cap S|}{|G_x|} = \sum_{g\in S} \...
- 66.8k
5
votes
1
answer
537
views
Which hyperbolic tilings are Cayley graphs?
I realise the question is easy but after asking to a few people (and never getting a clear answer), I thought it could be instructive to ask it here:
Given a regular tiling of the hyperbolic plane is ...
- 4,432
5
votes
1
answer
201
views
An Indepth Look at Isoperimetry in the Cayley Graph Generated by All Transpositions
Let $\Omega_n$ denote the symmetric/permutation group on $n$ objects.
Let $T_n \subseteq \Omega_n$ denote the set of transpositions.
Drop the $n$-subscripts.
Define the Cayley graph $G = (\Omega, E)$ ...
- 560
5
votes
1
answer
429
views
Cohomology of linear algebraic groups
Let $R$ be a commutative ring. Let $G\subset \mathrm{GL}_m$ be a linear algebraic subgroup. Has the group cohomology $H^i(G(R),R^m)$ been studied in the literature?
For example, do we know
(1) $H^...
user39380
5
votes
1
answer
309
views
Reductive groups over positive characteristics
Let $G$ be a connected split reductive group over a field $k$ of characteristic $p$. Let $\mathfrak{g}:=T_e(G)$ denote its Lie algebra. Let $T$ be a maximal split torus and $W$ the Weyl group (of the ...
- 2,751
5
votes
1
answer
217
views
Permutations of a group that are eventually left translations
$\DeclareMathOperator\FSym{FSym}\DeclareMathOperator\Sym{Sym}$Notation: for $X$ a set, $\Sym(X)$ the group of permutations of $X$, and let $\FSym(X)$ be the subgroup of finitely supported permutations ...
- 63.9k
5
votes
1
answer
184
views
Explicit short presentation of a 2-generated universal group?
A result of Higman states that there exists a finitely-presented group $G$ in which all other finitely-presented groups embed - I'll call such a group universal. Every countable group embeds in a 2-...
- 5,349
5
votes
1
answer
316
views
Connected permutation groups and wreath product
Let $G$ and $H$ be subgroups of the symmetric groups $\mathfrak S_m$ and $\mathfrak S_n$. Assume that $n>1$ and that $H$ is a 'connected' permutation group, that is, there is no non-trivial $H$-...
- 5,822
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
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