All Questions
Tagged with reference-request gr.group-theory
700 questions
3
votes
0
answers
231
views
What is known about "graph algebras"?
In lack for a better name I call a "graph algebra" a simple undirected graph $G=(V,E)$ and a binary mapping $+:E \rightarrow V$ such that:
(1) For all edges $(a,b)$ we have: $a+b \in N(a) \cap N(b)$, ...
user6671
3
votes
0
answers
170
views
What is $G_2(2^m)$, and how is it embedded in $\Gamma L_6(2^m)$?
I am trying to understand the classification of doubly transitive groups, specifically the nonsolvable affine case. Dixon and Mortimer (p.244) says there are three infinite families, one of which is $\...
- 7,633
3
votes
0
answers
62
views
Torus in the small Ree group ${}^2G_2$ over an infinite field
In “Simple group of Lie type” by R. W. Carter there is a remark (after Theorem 13.7.4):
It is not known whether $H^1$ coincides with the set of $\sigma$-invariant elements of $H$ if $\mathfrak{L}$ ...
- 3,186
3
votes
0
answers
365
views
Coinflation in cohomology
Let $U$ be a normal subgroup of a group $G$ of finite index. On cohomology, somewhat dual to the functorially defined restriction map, $\text{res}^G_U\colon H^n(G, A) \to H^n(U, A)$, the finite index ...
- 423
3
votes
0
answers
184
views
Mackey Obstruction Class with Integral Coefficients
Consider an exact sequence of groups
\begin{equation}
1\rightarrow H\rightarrow K\rightarrow G \rightarrow1~.
\end{equation}
Mackey theory enables us to understand representations of $K$ in terms of ...
- 2,097
3
votes
0
answers
269
views
Reference for the rank of the BN-pair of the finite simple groups of Lie type and not Chevalley
The rank of the BN-pair of a Chevalley group is the number of simple roots of its Lie algebra, which is the index of the name of its Dynkin diagram ($A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4,G_2 $).
...
3
votes
0
answers
113
views
Have locally principal crossed homomorphisms been studied?
Take a (multiplicative finite) group $H$ acting on the left (by automorphisms) on an (additive finite) abelian group $A$, and recall that the abelian (additive) group of crossed homomorphisms from $H$ ...
- 11.3k
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
3
votes
0
answers
494
views
Short exact sequences for amalgamated free products and HNN Extensions
I asked this question on math stackexchange (see here) but didn't get any answer so I thought I would post it here too:
If $A$ and $B$ are groups we have the following short exact sequence:
$$ 0 \to [...
- 721
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
3
votes
0
answers
102
views
Localized at $p$ integral representations of finite elementary $p$-groups
Let $C_p$ be a cyclic group of prime order $p$.
Let $F=C_p^n=C_p\times\dots\times C_p$ ($n$ times).
I would like to to classify finite dimensional representations of $F$ over ${\mathbb{Z}}$.
However, ...
- 14.2k
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 ...
3
votes
0
answers
209
views
Growth of the number of generators in hyperbolic groups
Let $G$ be an infinite hyperbolic group, and let us further assume that it is residually finite (or even LERF/GFERF) so that we have plenty of subgroups of finite index.
I would like to know if one ...
- 11.3k
3
votes
0
answers
222
views
torsion free for the 2nd cohomology group?
Let $G$ denotes an infinite coutable discrete group with Kazhdan's property (T),
My question is:
is it known that the 2nd cohomology group $H^2(G,\mathbb{Z}G)$ is torsion free?
Thanks in advance!
...
- 1,528
3
votes
0
answers
127
views
"A locally dual polar space for the Monster"
I am currently looking at Ronan and Stroth's 1984 paper Minimal Parabolic Geometries for the Sporadic Groups. When considering the $3$-minimal parabolic system of $F_{1}$, they cite a preprint by ...
- 295
3
votes
0
answers
156
views
Cancellations in products of two elements of a hyperbolic group
Let $G$ be a non-abelian free group with the standard generating set and the corresponding word metric. If we take two elements $g,h\in G$ and compute their product $gh$, some letters might cancel, ...
- 637
3
votes
0
answers
135
views
Groups acting on non-locally-finite trees with independence and specified local actions
Suppose I have a biregular tree $T_{m, n}$ (not necessarily locally finite), with distinct cardinal numbers $m, n$, so Aut$(T_{m, n})$ acts on $T_{m, n}$ without inversion. Let $V_m$ be those vertices ...
- 313
3
votes
0
answers
209
views
What is known about 2-modular representations of Ree groups of type $F_4$?
A too-vaguely worded question posted today about Suzuki and Ree groups reminds me to revisit a concern I never followed up years ago when assembling information about modular representations of finite ...
- 52.9k
3
votes
0
answers
153
views
On divisors occurring as subgroup sizes
Given a finite group $G$ define $D(G)$ to be the number of divisors $r$ of $|G|$ for which there exists a subgroup of $G$ of order $r$.
Clearly $D(G) \leq d(|G|)$, where $d(n)$ denotes the number of ...
- 731
3
votes
0
answers
144
views
Infinitely generated powerful pro-$p$ groups
A pro-$p$ group of finite subgroup rank has an open subgroup $P$ that is uniformly powerful, meaning that $[P,P]$ is contained in the group generated by $2p$-th powers in $P$, and raising elements to ...
- 4,728
3
votes
0
answers
264
views
How do I determine the smallest dimension of an irreducible $\mathbb{F}_p[G]$-module with a prescribed trivial fixed point space?
This is a crosspost from MSE since I haven't found an answer there yet.
I am not very familiar with modular representation theory or Brauer theory yet, however lately I have needed to use $\mathbb{F}...
- 1,173
2
votes
2
answers
594
views
Hausdorff Dimensions of Limit set of subgroups of SL(2,Z)
In a recent paper by Bourgain, Sarnak, Gamburd [1] talks about subgroups of $SL(2,\mathbb{Z})$.
Let $\Lambda$ be a finitely generated non-elementary subgroup of $SL(2,\mathbb{Z})$ with Hausdorff ...
- 22.8k
2
votes
2
answers
267
views
Irreducible representations of $G_4 = \langle a,b \mid a^{16}, b^{2}, baba^{-7}\rangle$ and other Semidihedral groups
I would like to know the irreducible representations of the group $G_4 = \langle a,b \mid a^{16}, b^2, baba^{-7}\rangle$ and its character table.
More than that, I would like to know the irreducible ...
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
2
votes
2
answers
419
views
Where can I find a table of the exponents of the sporadic groups?
Is there a table showing Sporadic Groups and their exponents, and, perhaps, other basic properties.
In particular, I'm interested in what the exponent of the Monster Group is. (Obviously the order is ...
- 373
2
votes
2
answers
1k
views
Magnus' embedding theorem
I am looking for a (preferably modern) reference to the following old result of Magnus.
Let $F$ be a free group of finite rank and
$$ F_1 = [F,F], F_2 = [F_1,F_1], \dots , F_{n+1} = [F_n,F_n], \dots ...
- 5,050
2
votes
2
answers
1k
views
Place stabilizers for the absolute Galois Group
Fix an algebraic closure, $\overline{\mathbb{Q}}$ for the rationals and consider the set, $B_p$, of all places of $\overline{\mathbb{Q}}$ over a fixed (possibly infinite) prime, $p$, of $\mathbb{Q}$. ...
- 1,049
2
votes
1
answer
233
views
Reference for surjectivity of the canonical map $R^{G_1} \otimes R^{G_2} \to R^{G_1 \cap G_2}$
Let $R$ be a commutative ring, $G$ a finite group with an action over $R$. Let $G_1, G_2 \subset G$ be two subgroups. Then the canonical map $R^{G_1} \otimes R^{G_2} \to R^{G_1 \cap G_2}$ is ...
- 23
2
votes
2
answers
381
views
Speed and absence of non-constant bounded harmonic functions
For a (symmetric) random walks on countable groups generated by $\mu$, there is a "brute-force computation" argument of Avez (1974) that shows that if the entropy $h_\mu$ is trivial then there are no ...
- 4,432
2
votes
1
answer
364
views
Totally aperiodic sequence
Let $A$ be a finite set. Let $A^k$ be the set of words in the alphabet $A$ of length $k$ and $A^*$ be the set of infinite words. I was looking for an element $a = \lbrace a_n \rbrace_{n \in \mathbb{N}}...
- 4,432
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
2
votes
2
answers
862
views
Non-split groups
I am looking for a reference with definitions on what it means for an algebraic group to be split, quasi-split, and non-split. I would like to see some examples of the different "types".
Thanks,
Tom
2
votes
2
answers
1k
views
description of functions of conditionally negative type on a group
Recall that a kernel conditionaly of negative type on a set $X$ is a map $\psi:X\times X\rightarrow\mathbb{R}$ with the following properties:
1) $\psi(x,x)=0$
2) $\psi(y,x)=\psi(x,y)$
3) for any ...
- 1,222
2
votes
1
answer
226
views
Name of the Marshall Hall paper in which he proved that the intersection of all subgroups of a fixed finite index is again finite index?
can someone please tell me? I couldn't find a reference in the paper I was reading.
- 87
2
votes
1
answer
111
views
Structure of elements of a finite group not contained in any conjugate of a proper subgroup
Let $G$ be a finite group and $H$ be a proper subgroup of $G$. It is elementary to prove that the union of all conjugates of $H$ under $G$,
$$U:=\bigcup_{\sigma\in G}\sigma^{-1}H\sigma,$$
is properly ...
2
votes
1
answer
210
views
Reference proving $\beta^{(2)}_1(G) \le d(G)-1$
I saw a paper that said it is well-known that for finitely generated group $G$: $\beta^{(2)}_1(G) \le d(G)-1$, but I can't find any reference proving it.
$d(G)$ denotes the minimal number of ...
- 197
2
votes
1
answer
514
views
Any representation is a subrepresentation of a direct sum of the regular representation
I need a reference for the following statement:
Let $G$ be a linear algebraic group over algebraically closed field $k.$ Let $V$ be a finite dimensional $G$-module. Then $V$ is subrepresentation of $...
- 661
2
votes
1
answer
217
views
A variation of closed-subgroup theorem
$\DeclareMathOperator\SO{SO}$Recall that the closed-subgroup theorem (Wikipedia link) says that a closed subgroup of a Lie group is a Lie group.
I am pretty sure that this theorem should have a "...
- 14.3k
2
votes
1
answer
231
views
A reference to the fact that a topologically transitive action of a group on a compact metrizable space has a dense orbit
I need a proper reference to the following obvious fact:
An action of a group $G$ on a nonempty compact metrizable space $K$ is topologically transitive (= the orbit $GU$ of any nonempty open set $...
- 41.9k
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
2
votes
1
answer
255
views
Does anyone have a copy of Salce's paper "Cotorsion theories for abelian groups"?
The paper "Cotorsion theories for abelian groups" by L. Salce, was published in 1979 in Symposia Math. 21, pages 1-21. According to Google Scholar, it's been cited 233 times, and I keep seeing ...
- 30.3k
2
votes
1
answer
197
views
Hall $\pi$ subgroups that controls its own fusion
Theorem:Let $P\in Syl_p(G)$ for a finite group $G$. Then $G$ has a normal $p-$ complement if and only if $P$ controls its own fusion.
I wonder if similar argument is true for Hall subgroups (in ...
- 1,169
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
2
votes
1
answer
253
views
Generalising right-angled Artin groups
An Artin group $G$ is determined by its Coxeter matrix $M$. This is a symmetric $n \times n$ matrix with entries from $\lbrace 2, 3, \ldots, \infty \rbrace$ that determine the relations between the ...
- 3,165
2
votes
1
answer
397
views
Counting Nearest Neighbors that Stay Nearest Neighbors after Random Rearrangements
Imagine we are making necklaces with $n$ beads, each bead is a different color from all others. Let's say we make one necklace. If we make another necklace with the same $n$ differently colored ...
- 161
2
votes
1
answer
129
views
Source for: a permutation group is multiplicity-free if and only if its 2-orbits define an association scheme
I have recently proven the following (at least, so I believe):
Theorem. Given a permutation group $\Sigma\subseteq\mathrm{Sym}(\Omega)$ on the set $\Omega:=\{1,...,n\}$, the following are equivalent:
...
- 13.6k
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 ...
2
votes
1
answer
465
views
The first Betti number of a finite covering space of a closed 3-manifold
Let $\tilde{M}\to M$ be a finite covering of closed 3-manifolds.
Is it possible that $\beta_1(\tilde{M})-1> [\pi_1(M):\pi_1(\tilde{M})](\beta_1(M)-1)>0$?
Here, $\beta_1(X)=\dim_{\mathbb{Q}}H_1(...
- 541
2
votes
1
answer
975
views
Explicit examples of Dehn presentations of hyperbolic groups
It is well known fact that a (f.g.) group is hyperbolic if and only if it admits a (finite) Dehn presentation.
My question concerns something I'm struggling with since the first time I read the proof ...
- 721
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