All Questions
Tagged with reference-request gr.group-theory
700 questions
3
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426
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Naturality of the transfer in group cohomology
Let $G$ be a (discrete) group and $H\le G$ a subgroup of finite index. Then there is a transfer map
$$tr\colon\thinspace H^\ast(H;M)\to H^\ast(G;M)
$$
in group cohomology, where $M$ is any $G$-module ...
- 35.9k
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
3
votes
1
answer
243
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Free Automorphisms
If $\varphi$ is an automorphism of $G = \langle x_1, \ldots, x_n; \mathbf{r}\rangle$ such that there exists an automorphism of $F(x_1, \ldots, x_n)$, $\overline{\varphi}$, with $$x_i\varphi=_G x_i\...
- 2,821
3
votes
1
answer
165
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When the fundamental group of subgraph of groups embeds?
Given a connected graph of groups $\mathcal G$ (where edge maps are embeddings), by a subgraph we mean a graph of groups obtain by omitting some vertices, some edges, and replacing the remaining ...
- 1,338
3
votes
1
answer
216
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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. ...
- 2,224
3
votes
1
answer
244
views
Finitely-generated conjugation action on a subgroup that is not normal... what is that?
If $H \lhd G$, then $G$ acts on $H$ by conjugation. I need to talk about this action but in a situation where $H$ is not (necessarily) normal. When $H \leq G$, there is a "partial action" of ...
- 6,652
3
votes
2
answers
676
views
Primitive action of wreath product
I say in advance that I am really new to Group Theory, so if my question is trivial I apologize in advance.
Let $A$ and $H$ be groups and $\Omega$ be a $H$-set. In this set-up, we can define the ...
- 1,252
3
votes
2
answers
469
views
Maximal size of minimal generating set
Let $G$ be a finite group. Denote by $D(G)$ the maximal size of a minimal generating set, (minimal in the sense of inclusion). I vaguely remember seeing recently something on $D(G)$. Can anyone refer ...
- 5,450
3
votes
1
answer
708
views
vanishing higher cohomology group for property T group?
Given a countable discrete group $G$ with Kazhdan's property (T), consider $\mathbb{C}G$ or $l^2(G)$ as a left $G$-module, then we can consider the group cohomology,
Is it known that $H^n(G, l^2(G))=...
- 1,528
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
3
votes
1
answer
213
views
Groups with special automorphism group
I am looking for all finite groups $G$ such that for each subgroup $H$ of $G$ and each automorphism $\sigma$ of $H$ there exists an automorphism $\psi$ of $G$ whose restriction to $H$ is $\sigma$. Is ...
- 792
3
votes
2
answers
337
views
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
3
votes
1
answer
375
views
Reference for triangle groups
Can anyone suggest to me some references for studying triangle groups? Especially the existence of finite index subgroups, subgroups isomorphic to fundamental groups of compact surfaces etc.
- 613
3
votes
2
answers
165
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References about the matrix generators of the finite subgroups of the orthogonal group O(4)
"On Quaternions and Octonions" by Conway and Smith gives the classification of the finite subgroups of the orthogonal group O(4). I want to get the explicit matrix generators of the finite subgroups. ...
- 31
3
votes
1
answer
273
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Preprint by Wall on Sjogren's theorem
In their account http://dx.doi.org/10.1016/0022-4049(87)90048-X of Sjogren's theorem, Cliff and Hartley refer to two articles:
[9] B. Hartley, A note on a lemma of Sjogren relating to. dimension ...
- 2,519
3
votes
1
answer
137
views
Pairs of elements in $F_n$ with distinct translation lengths
Let $F_n$ be a free group of rank n and consider all possible non-degenerate length functions on $F_n$.
Could I be directed to a reference that give two non-trivial non-power-conjugate elements $g,h \...
- 1,033
3
votes
1
answer
95
views
Question concerning Brauer's second main theorem, Brauer correspondent blocks and blocks covered by nilpotent blocks
A version of Brauer's second main theorem is as follows:
Let $G$ be a finite group, $x$ be a $p$-element of $G$, $B\in\mathcal{Bl}(G)$, and $\chi\in$ Irr$(B)$.
If $d_{\chi\mu}^x\neq 0$ and $\mu$ ...
- 1,755
3
votes
1
answer
226
views
Can MAGMA compute almost projective $kG$-homomorphisms?
Let $G$ be a finite group and $k$ be a finite field (big enough) whith char$(k)=p$ and $p\mid |G|$.
Let $M$ be a finitely generated $kG$-module.
We denote the first syzygy of $M$ by $\Omega(M)$, i.e....
- 1,755
3
votes
1
answer
118
views
On Groups of Maximal Class: Reference
I will be happy if one gives references (oncluding current research) for `classification' (structure) of $p$-groups of maximal class which contain abelian maximal subgroup (i.e. abelian subgroup of ...
- 1,169
3
votes
1
answer
149
views
Reference for decomposition in invariants and derived subgroup in a semidirect product of abelian groups
Let $A$ and $B$ be finite abelian groups with coprime order, and let $G=A\rtimes{}B$ be a semidirect product, via any action. Let $C\subseteq{}A$ be the subgroup of the elements of $A$ which are fixed ...
- 1,269
3
votes
1
answer
165
views
Reference request for statement concerning free subgroups of $ \mathrm{SL}_2(\mathbb{Z}). $
I am interested in finding a reference for the following claim:
There exists a free subgroup $F_2$ of $\mathrm{SL}_2(\mathbb{Z})$ on two generators that does not contain any nontrivial unipotent ...
- 131
3
votes
1
answer
224
views
Finiteness of a reflection group
Suppose that $V$ is a finite-dimensional real vector space and that $W\subseteq \operatorname{GL}(V)$
is a subgroup generated by reflections (elements $s$ of order $2$ whose locus of fixed points $H_s$...
- 3,137
3
votes
1
answer
141
views
The stabilizer of a pair of points in the acylindrically hyperbolic group is either finite or virtually cyclic
Given a group $G$, suppose $G$ admits a non-elementary acylindrical action
on a Gromov hyperbolic space $S$.
I heard that stabilizer of a pair of points on $\partial S$ in the acylindrically ...
- 199
3
votes
1
answer
302
views
Deciding isometry of unimodular lattices by Gram matrices
Say I have two unimodular lattices $A$ and $B$, represented by their Gram matrices.
Question: Is there an algorithm to decide whether $A$ and $B$ are isometric, i.e. whether there exists a matrix $S \...
- 9,501
3
votes
4
answers
654
views
A generalization of Landau's function
For a given $n > 0$ Landau's function is defined as $$g(n) := \max\{ \operatorname{lcm}(n_1, \ldots, n_k) \mid n = n_1 + \ldots + n_k \mbox{ for some $k$}\},$$
the least common multiple of all ...
- 798
3
votes
1
answer
319
views
Matrix transformation that "rotates" a matrix by $45^\circ$
I have an $n \times n$ integer matrix $A$. I want to obtain an $m \times m$ matrix $B$, where $m \ge n$, such that the rows of $A$ are the diagonals of $B$ and the columns of $A$ are the anti-...
- 2,993
3
votes
1
answer
372
views
Reference for real and complex projective representation of finite group
I'm not a mathematician. I've only learnt about irreducible representation of finite group, symmetric group and simple Lie group. In fact, I don't know projective representation belong to which part ...
- 977
3
votes
1
answer
309
views
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
3
votes
1
answer
137
views
Subalgebras with finite codimension
In group theory it is well-known that every subgroup of finite index contains a normal subgroup of finite index. It is not true in general that for Lie algebras every subalgebra of finite codimenslon ...
- 379
3
votes
0
answers
161
views
Generalized dimension property for rings
My question is very basic, I am looking for a characterization (and name) of rings $R$ satisfying the following property $\star$.
For any $V, W$ two finitely generated $R$-modules such that $V\oplus W\...
- 223
3
votes
0
answers
187
views
Bourgain-Gamburd-like theorems in the non-algebraic case
For $\mu$ a Borel probability measure on the compact group $G=\operatorname{SU}(d)$, Bourgain-Gamburd prove that the spectral radius of the associated operator on $L^2(G)$ is strictly less than one, ...
- 31
3
votes
0
answers
145
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
3
votes
0
answers
233
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 $\...
- 10.5k
3
votes
0
answers
463
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,...
- 613
3
votes
0
answers
115
views
Malliavin-Shavgulidze (type) measures on the group of measure-preserving invertible maps on $\mathbb T$?
The Malliavin-Shavgulidze measures on $\operatorname{Diff}^{1}(I)$ (with $I$ an interval of $\mathbb R$) are defined as the image $W_{\sigma} \circ f^{-1}$ of the Wiener measure $W_{\sigma}$ with ...
- 101
3
votes
0
answers
393
views
What about a Cayley n-complex for n>2?
Let $G$ be a finitely presented group. The Cayley graph of the finite generating set is a $1$-complex where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (...
3
votes
0
answers
205
views
Status of RFD groups and $C^*$-algebras
Motivated by this question and its great answers, I become very curious to know what do we know about RFD (residually finite dimensional) groups and $C^*$-algebras, e.g. do we know how these ...
- 1,586
3
votes
0
answers
296
views
Amenability, growth and asymptotic dimension
I recently found this question on MSE, relating growth of groups with whether they are amenable, elementary amenable or not. I would like to know if there is an extra relation to finite or infinite ...
- 267
3
votes
0
answers
153
views
Metropolis-Hastings sampling as a group action
Suppose that you have a topological space $\Omega \subset \mathbb R^n$ accompanied a measure $\mu$ and you're running an iterative sampling algorithm like Metropolis-Hastings. To sample you choose a ...
3
votes
0
answers
115
views
Reference for the Netto's theorem on the permutation groups which was mentioned in the paper of Frobenius
I'm trying to read 'Uber die Charaktere der mehrfach transitiven Gruppen' written by Frobenius.
There he mentioned some theorems of Netto.
I'm depending on the Google translator. and the translation ...
- 1,013
3
votes
0
answers
239
views
Groups with "just not" a property
There seems to be a standard trick in group theory which is to show that a group has a quotient group which "just not" has some property.
To make things clear:
let $\mathcal{P}$ be a group ...
- 4,432
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
3
votes
0
answers
58
views
Isoclinism for Lie groups: existing accounts of basic properties?
Philip Hall introduced the relation of isoclinism between two groups. One statement of the definition (not Hall's original statement) is to introduce a category whose objects are the canonical maps
$$...
- 25.8k
3
votes
0
answers
94
views
Clifford correspondence(s) from Fong-Reynolds theorem
The Fong-Reynolds theorem states a certain relationship between blocks of a normal subgroup $N\unlhd G$ and blocks of $G$, sometimes called the "Clifford correspondence for blocks". If one phrases ...
- 9,650
3
votes
0
answers
155
views
A variant on the Higman-Thompson groups
Let $C = \mathbb{Z}/d\mathbb{Z}$ ($d \ge 0$).
Let $D = \langle a_c : c \in C, t \mid a^2_c = t^d = 1, ta_ct^{-1} = a_{c+1} \rangle$.
let $E$ be the subgroup generated by $\{a_c : c \in C\}$ and let $...
- 4,728
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
3
votes
0
answers
136
views
Existence of loxodromic elements in certain subsets of $\text{PSL}_2(\mathbb C)$
Let $R$ be a subset of $\text{PSL}_2(\mathbb C)$ and consider its natural action on $\mathbb {CP}^1$. We say that $R$ is elementary if either $R$ is conjugated to a subset of $\text{SU(2)}$ or if ...
- 1,058
3
votes
0
answers
183
views
On Khelif's example of a group of countable cofinality having the Bergman property
A group $G$ is defined to have the Bergman property if for any subset $X=X^{-1}$ generating $G$ there exists $n$ such that $X^n=G$.
By a result of Bergman, the permutation group of any set has the ...
- 41.9k
3
votes
0
answers
217
views
References on a certain generalization of Dedekind groups
Recall that a group is called a Dedekind group if all of its subgroups
are normal. Also recall that a weaker property of a subgroup than normality
is that of being a TI-subgroup: a subgroup $H$ of a ...
- 19.6k