Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,182 questions
17
votes
1
answer
650
views
Cohomology of braid groups with coefficients in the group ring
Let $\mathbf B_n$ be the braid group on $n$ strings.
What is known about the cohomology of $\mathbf B_n$ with coefficients in its integral group ring: $H^*(\mathbf B_n;\mathbb Z \mathbf B_n)$?
- 1,782
17
votes
1
answer
3k
views
Do these conditions on a semigroup define a group?
As is well known, if $S$ is a semigroup in which the equations $a=bx$ and $a=yb$ have solutions for all $a$ and $b$, then $S$ is a group. This question arose when someone misunderstood the conditions ...
- 7,217
17
votes
1
answer
1k
views
Does O'Nan-Scott depend on CFSG?
My question is in the title. Some context: there are two versions of the O'Nan-Scott theorem. The first, weaker version, is due to O'Nan and Scott (independently) and gives the structure of the ...
- 11.2k
17
votes
1
answer
927
views
Do there exist non-isomorphic groups with the same cohomology?
For any group $G$, cohomology can be viewed as a functor
$$
H^\ast(G,-): G{\sf\text{-}mod}\to {\sf GrAbGrp},
$$
where $G{\sf\text{-}mod}$ denotes the category of (left) $\mathbb{Z}[G]$-modules and ${\...
- 35.9k
17
votes
3
answers
1k
views
The second homotopy group of a simple CW-complex
Let $X$ be a CW-complex with
one 0-cell
two 1-cells
three 2-cells
no cells in dimensions 3 or higher.
Is it always true that $\pi_2(X)\ne 1$?
- 1,181
17
votes
1
answer
875
views
Which finitely presented groups can be distinguished by decidable properties?
This question continues the line of inquiry
of these
three
questions.
Question. Which finitely presented groups can be
distinguished by decidable properties?
To be precise, let us say that φ is ...
- 236k
17
votes
1
answer
863
views
Inner? Automorphisms
It is a fairly easy result of Edna Grossman's that any automorphism of a (finitely generated) free group which acts by conjugation on every primitive element ( primitive element in$F_n$ is one which ...
- 96.4k
17
votes
1
answer
765
views
Are groups with every proper, non-trivial subgroup infinite cyclic simple?
In the 1970s Ol'shanskii constructed a non-cyclic finitely generated group $G$ with the following properties:
Every proper, non-trivial subgroup of $G$ is infinite cyclic.
If $X^m=Y^n$ for $X, Y\in G$...
- 2,821
17
votes
1
answer
1k
views
Why do these two Monster-related calculations yield $163$?
Fact 1: (1979, Conway and Norton)$^{1}$
"There are $194-22-9=\color{blue}{163\,}$ $\mathbb{Z}$-independent McKay-Thompson series for the Monster."
Note: There are 194 (linear) irreducible ...
- 12.6k
17
votes
1
answer
701
views
Outer automorphisms of finite extensions
Let $H$ be a finite index subgroup of a finitely generated group $G$. Assume that $Out(H)$ is finite. Can $Out(G)$ be infinite?
- 11.1k
17
votes
1
answer
1k
views
About an exercise in Serre's "Trees"
Following section 1.4, which is entitled "Constructions Using Amalgams" is the innocent-enough looking exercise:
Show that the group defined by the presentation (presumably on the generators $x_1, ...
- 273
17
votes
1
answer
448
views
The possible degrees of $\mathbb{Q}(a,b)$ in terms of the degrees of $a$ and $b$
Given two positive integers $n,m$, which positive integers $d$ appear as the degree of $\mathbb{Q}(a,b)$ for two algebraic numbers $a$ and $b$ of degrees $n$ resp. $m$?
Two necessary conditions are $\...
- 5,482
17
votes
1
answer
420
views
Freeness of tensor product
Let $G$ be a finite group. Is $\mathbb{Z}G\otimes_{Z(\mathbb{Z}G)}\mathbb{Z}G$ free as a $\mathbb{Z}$-module, where $Z$ denotes the centre?
- 355
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
$$
\delta : C^k \to C^{k+1}
$$
$$
(\delta f)(g_1, \ldots, ...
- 12.8k
17
votes
2
answers
513
views
On a special type of normed linear spaces
Let $(V,\|.\|)$ be a normed linear space such that for every group $(G,*)$, every function $f:G \to V$ satisfying
$$
\|f(x*y)\|\ge \|f(x)+f(y)\|,\qquad\forall x,y\in G,\tag{Z}
$$ is a group ...
- 1,209
17
votes
1
answer
1k
views
Is every normal subgroup of $\mathrm{SL}_2(\mathbb{Z}/n)$ also normal inside $\mathrm{GL}_2(\mathbb{Z}/n)$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Is every normal subgroup of $\SL_2(\mathbb{Z}/n)$ also normal inside $\GL_2(\mathbb{Z}/n)$?
Of course, it suffices to ask the question when $n = ...
- 7,527
17
votes
2
answers
2k
views
Non-isomorphic groups with the same oriented Cayley graph
There are many examples of two non-isomorphic groups with the same Cayley graph. If the graph is non-oriented, asking for the generating set to be minimal does not make the task much harder. However, ...
17
votes
1
answer
1k
views
Epimorphisms $\mathbb{Z}^{\mathbb{N}} \to \mathbb{Z}^{\mathbb{N}}$ are split
Consider the additive group of integer sequences $\mathbb{Z}^{\mathbb{N}}$. Why does every epimorphism of groups $\mathbb{Z}^{\mathbb{N}} \to \mathbb{Z}^{\mathbb{N}}$ split? $(\star)$
Actually this ...
- 63.1k
17
votes
1
answer
798
views
Are There Always Group Generators Which Give Unimodal Growth?
Suppose $G$ is a $k$-generated finite group. Is there always a set of $k$ elements which generate the group and have a unimodal counting function?
Background:
The counting function, $f(n)$, is a ...
- 2,143
17
votes
1
answer
656
views
Groups with finitely generated center
Does every group with a finite classifying space have finitely generated center?
Remarks:
If $G$ is a finitely generated group with infinitely generated center $Z(G)$,
then the quotient $G/Z(G)$ is ...
- 29.1k
17
votes
0
answers
692
views
Monstrous Langlands-McKay or what is bijection between conjugacy classes and irreducible representation for sporadic simple groups?
Context: The number of conjugacy classes equals to the number of irreducuble representations (over C) for any finite group. Moreover for the symmetric group and some other groups there is "good ...
- 24.9k
17
votes
0
answers
536
views
Question about combinatorics on words
Let $\{a_1,a_2,...,a_n\}$ be an alphabet and let $\{u_1,...,u_n\}$ be words in this alphabet, and $a_i\mapsto u_i$ be a substitution $\phi$.
Question: Is there an algorithm to check if for some $m,k$...
user6976
17
votes
0
answers
824
views
What's the big deal about $M_{13}$?
$M_{13}$ is the Mathieu groupoid defined by Conway in
Conway, J. H. $M_{13}$. Surveys in combinatorics, 1997 (London), 1–11,
London Math. Soc. Lecture Note Ser., 241, Cambridge Univ. Press, ...
- 11.2k
17
votes
0
answers
513
views
Maximum automorphism group for a 3-connected cubic graph
The following arose as a side issue in a project on graph reconstruction.
Problem: Let $a(n)$ be the greatest order of the automorphism group of a 3-connected cubic graph with $n$ vertices. Find a ...
- 37.7k
17
votes
0
answers
449
views
Splay trees and Thompson's group $F$
( I apologize for only indicating some easy to find references, but new users are not allowed to link more than five). This is very speculative, but:
Question: Is there a reformulation of the Dynamic ...
- 992
17
votes
0
answers
969
views
Groups generated by 3 involutions
Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition
$\tau_{r_1(m_1),r_2(m_2)}$ be the ...
- 19.6k
16
votes
7
answers
2k
views
two conjugate subgroups and one is a proper subset of the other? plus, a covering space interpretation.
Recently I've been reading J.P. May's A Concise Course in Algebraic Topology. In the section on the classification of covering groupoids, he mentions that sometimes a group G may have two conjugate ...
- 6,069
16
votes
4
answers
4k
views
Is there an infinite group with exactly two conjugacy classes?
Is there an infinite group with exactly two conjugacy classes?
- 343
16
votes
2
answers
1k
views
Groups containing each other as finite index subgroups
Suppose that $G, H$ are finitely generated groups such that $H$ is isomorphic to a finite index subgroup of $G$ and vice versa. Does it follow that $G$ is isomorphic to $H$?
I am sure that the ...
- 12.3k
16
votes
5
answers
6k
views
Finite subgroups of ${\rm SL}_2(\mathbb{Z})$ (reference request)
I recently read the statement "up to conjugacy there are 4 nontrivial finite subgroups of ${\rm SL}_2(\mathbb{Z})$." They are generated by
$$\left(\begin{array}{cc} -1&0 \\\ 0&-1\end{array}\...
- 2,869
16
votes
3
answers
3k
views
If the center of a finite group is trivial, are there two elements whose centralizers intersect trivially?
Let $G$ be a (discrete) group. Define $k^*(G)$ as the minimal cardinality of a set $S \subset G$ such that $C_G(S) = Z(G)$. Define $k(G) = k^*(G)$ if $G$ has trivial center (i.e. $|Z(G)| = 1$), and $k(...
- 6,652
16
votes
3
answers
654
views
Is the isomorphism problem for amenable groups decidable?
Is it algorithmically decidable if two finitely presented amenable groups are isomorphic?
Or slightly different:
Does there exist a family of amenable groups (indexed by natural numbers) for which ...
- 261
16
votes
4
answers
905
views
Groups that satisfy ${ [x,y]^2 \approx 1 }$
Lately, I have been constructing finite involution monoids that generate varieties with $2^{\aleph_0}$ subvarieties. One construction requires groups that violate the identity ${ [x,y]^2 \approx 1 }$, ...
- 563
16
votes
3
answers
2k
views
The advantage of asymmetric objects
We know that it is usually much easier to work with highly symmetry objects, the objects that have many automorphisms like the sphere, Lie groups, complete graph,... But is there any advantage of ...
- 1,462
16
votes
3
answers
2k
views
What are the main open problems in the theory of amenability of groups?
I have read the Paterson and Runde books about amenability of groups, but I do not know what are the most intriguing questions in this area today.
A survey or a list of questions would be welcome.
16
votes
2
answers
818
views
Decomposing $(\mathbb C^n)^{\otimes m}$ as a representation of $S_n\times S_m$
$V=\mathbb C^n$ is a $\mathbb CS_n$-module, where $S_n$ is the symmetric group of degree $n$, via the representation sending a permutation to the corresponding permutation matrix. The tensor power $V^...
- 38.6k
16
votes
2
answers
1k
views
The number of group elements whose squares lie in a given subgroup
This number is divisible by the order of the subgroup http://arxiv.org/abs/1205.2824.
The proof is short but non-trivial. Is this fact new or is it known for a long time?
- 3,932
16
votes
3
answers
1k
views
Cohomological dimension of $G \times G$
$\DeclareMathOperator\cd{cd}$A question that I have already posted in the Mathematics section, but which seems to be too delicate for that section (see here and here):
Let $\cd(G)$ denote the ...
- 843
16
votes
3
answers
3k
views
Are there other semidirect product/crossed products in other areas
Suppose $(O, G, \alpha)$ is a triple where $O$ is some mathematical object, $G$ is a group and $\alpha : G \rightarrow Aut(O)$. Many different areas of mathematics study such triples. However, I only ...
- 3,984
16
votes
2
answers
2k
views
Status of Quillen's conjecture on elementary abelian p-groups
These are questions on D. Quillen's 1978 paper Homotopy properties of the poset of nontrivial p-subgroups of a group.
Let $G$ be a finite group, $p$ a prime number, $\mathcal S(G)$ the poset of non-...
- 2,734
16
votes
2
answers
889
views
Why are Thompson's groups called $F$, $T$ and $V$?
Why are Thompson's groups called $F$, $T$ and $V$?
I never saw Thompson's unpublished notes, in which he introduces these groups; maybe an explanation can be found there?
- 8,401
16
votes
4
answers
1k
views
Origin of group theory problem (bound on number of Sylow subgroups)
This problem (prove that the number of Sylow subgroups of a finite group $G$ is bounded by $\frac{2}{3}|G|$) posted on MSE proved rather difficult to solve. The OP has been silent about where the ...
- 2,168
16
votes
3
answers
1k
views
Torsion subgroups of hyperbolic groups are finite?
Is it true that torsion subgroups of hyperbolic groups are finite?
I have a vague memory that this is true, perhaps due to Ol'shanskii, but have been struggling to find a reference.
(By a torsion ...
- 161
16
votes
2
answers
2k
views
Algorithms in hyperbolic groups
I'm stuck in some algorithms in hyperbolic groups, which may be rather simple.
Let $G$ be a hyperbolic group given by a finite presentation. It is known that the hyperbolicity constant $\delta$ can ...
- 619
16
votes
3
answers
797
views
For which rings R is SL_n(R) generated by its n-1 fundamental copies of SL_2(R)?
By "fundamental copies" of $SL_2(R)$ in $SL_n(R)$, I mean those embedded along the diagonal (for instance, if $n=3$, those are the upper left and lower right corner copies of $SL_2(R)$ embedded in $...
- 563
16
votes
2
answers
5k
views
How to compute all irreducible representations of a finite group ? (how GAP is doing this?)
Let us "take" a finite group G. Here "take" I mean any type of group-theoretic description you prefer: e.g. as an explicit subset of GL (or other group) or Cayley table, whatever.
Question: How ...
- 24.9k
16
votes
2
answers
880
views
Groups which maintain all their subgroups’ automorphisms as inner automorphisms
Are there any groups, finite or infinite, other than the first three symmetric groups which maintain all their subgroups’ automorphisms as inner automorphisms (every automorphism of every subgroup ...
- 2,773
16
votes
3
answers
715
views
Group with non-trivial center containing trivially-intersecting copies of itself
I'm trying to think of an example of a group $G$ with non-trivial center such that there exist subgroups $H_1,H_2\le G$ both isomorphic to $G$ and satisfying $H_1\cap H_2=\{1\}$. Does such a group ...
- 3,740
16
votes
2
answers
626
views
Cantor-Bernstein for quasi-isometric embeddings?
Suppose that two finitely generated groups quasi-isometrically embed into each other. Does it follow that the two groups are quasi-isometric? Recall that a quasi-isometry is a quasi-isometric ...
- 11.1k
16
votes
1
answer
955
views
Finitely generated group with $\aleph_0<X_G<2^{\aleph_0}$ normal subgroups?
Let $X_G$ be the number of normal subgroups of a group $G$. Are there examples of finitely generated groups $G$ where it is consistent to have $\aleph_0<X_G<2^{\aleph_0}$ normal subgroups? Also ...
user35370