All Questions
Tagged with gr.group-theory reference-request
661
questions
7
votes
0
answers
130
views
An identity for characters of the symmetric group
I am looking for a reference for the identity
$$\chi_\lambda(C)=\frac{\dim(V_\lambda)}{|C|}\sum_{p\in P_\lambda,\,q\in Q_\lambda,\,pq\in C}\operatorname{sgn}(q)$$
for the irreducible characters of the ...
- 1,514
17
votes
1
answer
1k
views
Explicit character tables of non-existent finite simple groups
In connection with the historical development of the classification of finite simple groups, I am interested in a particular aspect that seems to be less well-documented than the main narrative of ...
- 25.5k
2
votes
0
answers
75
views
Hereditarily just-infinite pro-$2$ groups
An infinite profinite group $G$ is called just-infinite if all non-trivial closed normal subgroups of $G$ have finite index. A profinite group is called hereditarily just-infinite if every open ...
- 499
2
votes
0
answers
127
views
$p$-adic Banach group algebra
Let $G$ be a discrete group. Consider the Banach $\mathbb{Z}_p$-algebra: $$c_0(G, \mathbb{Z}_p) = \{ F : G \to \mathbb{Z}_p \mid \lim_{g \to \infty} |F(g)|_p = 0 \}$$ with the product given by the ...
- 1,117
6
votes
1
answer
178
views
Classification of algebraic groups of the types $^1\! A_{n-1}$ and $^2\! A_{n-1}$
This seemingly elementary question was asked in Mathematics StackExchange.com: https://math.stackexchange.com/q/4779592/37763.
It got upvotes, but no answers or comments, and so I ask it here.
Let $G$ ...
- 13.4k
1
vote
1
answer
194
views
Conditions of P for existence of orthogonal matrix Q and permutation matrix U satisfying QP = PU
Question: Let $P\in \mathbb{R}^{d\times n}$ be a $d$-rank real matrix and $PP^T = c I_d$ with a certain constant $c > 0$. Under what additional conditions of $P$ does there exist an orthogonal ...
- 187
9
votes
1
answer
670
views
Where has this structure been observed?
$\newcommand{\M}{\mathcal{M}}$Let $M$ be a monoid. Consider the following structure:
$R_X,R_Y:\mathbb{Z}^2 \to M$ satisfying the following "compatiblity-relation":
$$R_X (x, y) \cdot R_Y (x +...
- 6,581
2
votes
0
answers
77
views
Simple modules and trivial source modules
Let $(K,\mathcal{O},k)$ be a large enough $p$-modular system.
In this context, I would like to ask what is known about the following question:
when are simple $kG$-modules trivial source modules?
So ...
- 1,685
16
votes
1
answer
372
views
Is there a relationship between Broué's abelian defect group conjecture and Alperin's weight conjecture?
Let $G$ be a finite group, let $k$ be a large enough field of characteristic $p>0$. Let $p\mid |G|$.
Broué's abelian defect group conjecture states the following:
Let $B$ be a block of $kG$ with
...
- 1,685
1
vote
0
answers
74
views
$p$-modular splitting systems and the characteristic of the ring $\mathcal{O}$
Let $k=\overline{k}$ be a field of characteristic $p$.
Let $(K,\mathcal{O},k)$ be a $p$-modular system.
Let both $k$ and $K$ be splitting fields for $G$ and its subgroups.
The ring $\mathcal{O}$ can ...
- 311
0
votes
0
answers
115
views
normalizer info for subgroups
In [1], Griess classified the maximal nontoral elementary abelian subgroups of algebraic groups. For the exceptional types, normalizer info was also given. Is there any work out there providing ...
- 501
6
votes
2
answers
349
views
Twisted forms with real points of a real Grassmannian
Let $X={\rm Gr}_{n,k,{\Bbb R}}$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb R}^n$.
We regard $X$ as an ${\Bbb R}$-variety with the set of complex points $X({\Bbb C})={\rm Gr}_{n,k,{\...
- 13.4k
14
votes
2
answers
548
views
Existence of a regular semisimple element over $\mathbb{F}_{q}$
This is probably old, a Chevalley level of old, but I'm not at all an expert in this field so I need help.
Let $G$ be a simply connected (almost) simple linear algebraic group defined over $K=\mathbb{...
- 435
1
vote
0
answers
83
views
Existence of countable dense normal subgroups of global Galois group
Let $K$ be a number field, $S$ a finite set of primes of $K$ and $ G_{K,S} $ the Galois group of the maximal extension of $ K $ (inside a fixed algebraic closure of $K$) unramified outside $ S $. In ...
- 755
6
votes
0
answers
217
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) $$
is a group ...
- 6,454
5
votes
1
answer
228
views
Product of all conjugacy classes
Related to this post of my coauthor Sebastien, I should also mention that one can also prove the following dual result:
For any finite group G, the following identity holds:
$$
\left(\prod_{j=0}^m \...
- 837
9
votes
1
answer
226
views
Original references for the Hall - Witt identity
The group identity
$$
[[a,b^{-1}],c]^b \cdot [[b,c^{-1}],a]^c \cdot [[c,a^{-1}],b]^a = 1
$$
is commonly attributed to Hall and Witt (here $x^y:=y^{-1}xy$ and $[x,y]:=x^{-1}y^{-1}xy$). However, ...
- 16.5k
5
votes
0
answers
273
views
Adjoint identity on finite nilpotent groups
Let $G$ be a finite nilpotent group. Consider the following (adjoint) identity from [BuPa, Theorem 9.6]:
$$\left(\prod_{\chi \in \mathrm{Irr}(G)} \frac{\chi}{\chi(1)} \right)^2 =\frac{|Z(G)|}{|G|}\...
- 25.5k
1
vote
0
answers
105
views
List of automorphism groups of low-dimensional complex commutative algebras?
Let $\mathcal{A}$ be a finite-dimensional commutative associative unital $\mathbb{C}$-algebra. I am looking for a list (of further examples of) $\operatorname{Aut}_\mathbb{C}(\mathcal{A})$, the group ...
- 6,643
4
votes
2
answers
355
views
Splitting field for $\mathrm{GL}(2,p)$ - reference request
It seems to me from a quick glance at several sources describing the complex and modular irreducible representations of $\mathrm{GL}(2,p)$ that any field $K$ containing a primitive $(p-1)$-root of ...
- 36.8k
3
votes
0
answers
115
views
Reference showing no proper subgroups of p-adic orthogonal groups surject onto mod p orthogonal groups
I am looking for a reference for the following statement:
Let $O$ be an orthogonal group associated to a nondegenerate quadratic form of rank $r$ over the p-adic integers $\mathbb Z_p$. Suppose $r$ is ...
- 1,308
1
vote
0
answers
114
views
Multiplicativity of Euler–Poincaré characteristics of cohomology of pro-$p$ groups
While reading a paper, I found a mentioning that for an extension $1 \rightarrow H \rightarrow G \rightarrow N \rightarrow 1$ of pro-$p$ groups, the Euler–Poincaré characteristics $\chi(H)$, $\chi(G)$,...
- 1,043
1
vote
0
answers
142
views
Units in group rings in SAGE
Is there a recorded/known SAGE code to compute units in integral group rings for finite abelian groups ?
I would be happy with a code that only works for cyclic groups. I sort of know how to ...
- 12.6k
9
votes
1
answer
332
views
The maximum order of torsion elements in ${\rm GL}_n(\mathbb{Z}_p)$ or ${\rm GL}_n(\mathbb{F}_p[[T]])$
This question is inspired by Upper bound on order of finite subgroups of GL_n(Z_p)?. It's showed that the supremum of orders of finite subgroups of ${\rm GL}_n(\mathbb{Z}_p)$ is finite and can be ...
- 755
3
votes
1
answer
203
views
Passing to normal forms in graphs of groups
Given a word $w \in X^{\pm 1}$ representing an element of the free group $F(X)$ there is a (usually non-unique) sequence $w=w_0 \to w_1 \to \cdots \to w_r$ with $|w_i|>|w_{i+1}|$ where $w_r$ is the ...
- 963
3
votes
0
answers
206
views
A bridge between the algebraic / differential geometry of $\frak{sl}_2(\mathbb{C})$ and the Sheffer-Appell calculus and combinatorics
In "Four examples of Beilinson-Bernstein localization", Anna Romanov introduces the basis
$m_k = \frac{(-1)^k}{k!} \partial^k \delta $
on p. 9, where $\partial$ is a partial derivative and $\...
- 9,407
6
votes
1
answer
534
views
If G is an almost simple group, then Aut(G) is complete?
If G is an almost simple group, then Aut(G) is complete?
Apologies - I meant to post this on Stack Exchange
Just wondering if anyone has a reference to the above - it's quoted on Wikipedia (so ...
- 63
9
votes
2
answers
568
views
Finite subgroups of $\mathrm{SO}(n)$ and $\mathrm{O}(n)$
Question 1:Is there a reference that lists all possible finite subgroups and their orders of $\mathrm{SO}(n)$ and $\mathrm{O}(n)$ for $n=4$ or even higher $n$ over the real numbers?
I can only find ...
- 25.4k
2
votes
0
answers
344
views
Conceptual proof of fundamental theorem of finite abelian groups
I'm looking for a conceptual proof of the following statement:
Lemma: Let $G$ be a finite abelian $p$-group. Let $a$ be an element of maximal order. Then $G=\langle a \rangle \times H$ for some ...
- 2,519
9
votes
2
answers
630
views
Torsion-free virtually free-by-cyclic groups
Is it known if there are any examples of a finitely generated group $G$ such that:
$G$ has a finite index subgroup $H$ which is free-by-cyclic
$G$ itself is not free-by-cyclic
$G$ is torsion-free
...
- 283
3
votes
1
answer
191
views
Identifying group extension from cohomology class of $D_8$
I have the following problem. It is well known that $H^\ast(D_8,\mathbb{Z}/2)\cong \mathbb{F}_2[x,y,w]/(xy=0)$ with $|x|=|y|=1$ and $|w|=2$ (see Adem,Milgram "Cohomology of finite groups"). ...
- 1,679
1
vote
1
answer
86
views
Quotients of pro-$p$ groups linear over a complete Noetherian local ring
Let $R$ be complete Noetherian local ring with finite residue field $\mathbb{F}$ of characteristic $ p $. We say that a pro-$p$ group $G$ is linear over $R$ if it is isomorphic to a closed subgroup of ...
- 755
1
vote
0
answers
72
views
Bottleneck edge in lattice of subgroups
Let $G$ be a finite group. Define the bottleneck weight of a chain of subgroups $$\operatorname{id}=H_0 < H_1 < \ldots < H_n = G$$ to be the maximum value over the indices $[H_{i+1} : H_i]$ ...
- 378
4
votes
0
answers
75
views
Duality for finite quotient groups of finitely generated free abelian groups
$\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\Hom}{{\rm Hom}}
$ The following lemma is certainly known.
Lemma (well-known).
Let $B$ be a lattice (that is, a finitely generated ...
- 13.4k
1
vote
0
answers
83
views
A duality of finite groups coming from a surjective homomorphism with finite kernel of algebraic tori
$\newcommand{\Hom}{{\rm Hom}}
\newcommand{\Gm}{{{\mathbb G}_{m,{\Bbb C}}}}
\newcommand{\X}{{\sf X}}
$ I am looking for a reference for the following lemma (for which I know a proof):
Lemma.
Let $\...
- 13.4k
4
votes
1
answer
255
views
Is a Lie subgroup whose center is closed, a closed subgroup itself?
I want to show that a certain Lie subgroup (i.e. generated by the exponential of elements in some Lie subalgebra) of a Lie group is closed. My knowledge of the subject of Lie groups is rudimentary, ...
- 4,174
9
votes
1
answer
277
views
Comparing cohomology of a total complex with the cohomology of semidirect product
$\DeclareMathOperator{\Tot}{Tot}$I have the following problem. Let $H$ and $G$ be groups such that $H$ acts on $G$, i.e., there exists a group homomorphism $H\to \mathrm{Aut}(G)$ and let $M$ be an ...
- 1,679
9
votes
1
answer
211
views
Yang-Mills algebra and lower central series of surface groups
Here is a connection that I recently noticed, but I haven't quite been able to make sense of. It might follow from well-known facts; apologies, if so. This is quite far from my area.
First, in "...
12
votes
2
answers
503
views
Moments of character degrees - is this result new or folklore?
Context
$\DeclareMathOperator\cp{cp}\DeclareMathOperator\AM{AM}\DeclareMathOperator\A{A}$For a finite group $G$ and $k\in\mathbb R$, define
$$
m_k(G) = \frac{1}{|G|} \sum_{\pi\in\widehat{G}} (d_\pi)^{...
- 25.3k
3
votes
1
answer
198
views
Reference request: Serre's Groupes discrets
I'm reading some articles and at some point they both reference:
J-P. Serre: Groupes discrets (in collaboration with H. Bass),
Collège de France, 1969
However I have trouble finding this reference. ...
- 2,175
3
votes
0
answers
144
views
Automorphism-conjugacy
If $G$ is a group, we can say $g$ is automorphism-conjugate to $f$ if there is a group automorphism $\alpha : G \to G$ such that $g = \alpha(f)$. This is an equivalence relation.
Is there a standard ...
- 6,327
1
vote
1
answer
229
views
A group-theoretic lemma in a paper by Ershov and He
In the proof of Lemma 2.1 in
Ershov, Mikhail; He, Sue, On finiteness properties of the Johnson filtrations, ZBL06904638,
the authors claim the following (without proof).
Let $G$ be a finitely ...
- 1,307
3
votes
0
answers
309
views
Representations of triangle groups
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PSL{PSL}$I am self-studying triangle groups and the following question comes up.
Let $G$ denotes $(2,3,7)$ triangle group. It is symmetry group of $(2,...
- 321
2
votes
0
answers
92
views
Question concerning relationships between different $p$-modular systems and Brauer character values
Let $(K,\mathcal{O},k)$ be a large enough $p$-modular system, where $\mathcal{O}$ is a complete discrete valuation ring of characteristic zero with unique maximal ideal $J(\mathcal{O})$, algebraically ...
- 1,685
10
votes
3
answers
1k
views
Axioms for the category of groups
Certain categories of mathematical structures have had synthetic axiom systems developed for them. One particularly well known such category is the category of sets and functions $\mathit{Set}$, which ...
- 2,016
4
votes
1
answer
236
views
Wedderburn decomposition of special linear groups
$\DeclareMathOperator\SL{SL}\newcommand\card[1]{\lvert#1\rvert}$I want to study about Wedderburn decomposition of group algebra $k\SL(n,\mathbb{F}_p)$ where $k$ is either an algebraically closed field ...
- 239
18
votes
2
answers
825
views
Has anyone seen this construction of the Weil representation of $\mathrm{Sp}_{2k}(\mathbb{F}_p)$?
$\def\FF{\mathbb{F}}\def\CC{\mathbb{C}}\def\QQ{\mathbb{Q}}\def\Sp{\text{Sp}}\def\SL{\text{SL}}\def\GL{\text{GL}}\def\PGL{\text{PGL}}$Let $p$ be an odd prime. The Weil representation is a $p^k$-...
- 150k
8
votes
1
answer
282
views
"Novelty" maximal subgroups in $S_n$
What are the maximal subgroups $M < S_n$ such that $M \cap A_n$ is not maximal in $A_n$?
Maximal subgroups of $S_n$ are described by the O'Nan-Scott theorem and very extensively studied in many ...
- 2,771
2
votes
1
answer
384
views
Growth rate of an outer automorphism of a free product
$\DeclareMathOperator\Out{Out}$Let $G=G_1\ast\cdots\ast G_k\ast F_p$ be a Grushko decomposition of a finitely generated group $G$, $\mathcal{O}$ the outer space relative to this decomposition, $[\phi]\...
- 237
9
votes
0
answers
258
views
Is “simplicity is elementary” still hard? (Felgner’s 1990 theorem on simple groups, and subsequent work)
I came across a reference in this MathOverflow answer to an intriguing result of Ulrich Felgner [1]: among finite non-Abelian groups, the property of being simple is first-order definable. According ...
- 18.8k