Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
1,095 questions
7
votes
1
answer
707
views
The Higman group II
This question is related to the question The Higman group (with a nice answer by M. Sapir). So for background, please,
see the above cited question.
The Higman group has an automorphism $h(a_j)=a_{j+...
- 723
7
votes
0
answers
378
views
General solution of the Multiplicative symmetry equation $f(xf(y))=f(f(x)y)$ in nonabelian groups
As we know, the functional equation $f(xf(y))=f(f(x)y)$ was completely solved in abelian groups (by J. G. Dhombres, Solution... $f(x\ast
f(y))=f(y\ast f(x))$, Aequationes Math. 15 (1977), 173--193, ...
- 995
7
votes
2
answers
1k
views
Abelianization of Lie groups
If G is a group, its abelianization is the abelian group A and the map G → A such that any map G → B with B abelian factors through A. Abelianization is a functor, and in general a very ...
- 54.6k
7
votes
2
answers
1k
views
Malcev Lie algebra and associated graded Lie algebra
Suppose $L$ is a nilpotent finite-dimensional Lie algebra over $\mathbb{Q}$ of class $c$. We can define an associated graded Lie algebra to $L$ that, as a vector space, is:
$$\bigoplus_{i=1}^c \...
- 7,320
7
votes
2
answers
456
views
Subgroup property stronger than being characteristic
In what follows, all groups are assumed to be finite.
Recall that if $K \leq H \leq G$ are groups, $K$ is said to be a weakly closed subgroup of $H$ in $G$ if, for all $g \in G$, $g^{-1}Kg \leq H$ ...
- 3,645
7
votes
4
answers
1k
views
Consequences of the Inverse Galois Problem
Are there any papers written about the consequences of the Inverse Galois Problem in case it is proved to be true or false?
We know a lot of things that would be true if the Riemann Hypothesis holds. ...
7
votes
2
answers
384
views
Transitive actions of $Aut(F_2)$ on surjections from $F_2\twoheadrightarrow G$
Here $F_2$ is the free group on two generators $x,y$. I'm interested in examples of finite groups $G$ such that $Aut(F_2)$ acts transitively on the set of surjections $F_2\rightarrow G$. (In ...
- 10.7k
7
votes
2
answers
586
views
Rotation numbers for amenable group actions on the circle
Given an orientation-preserving homeomorphism $f: S^1 \to S^1$, one can define its rotation number $\rho(f) \in \mathbb{R}/\mathbb{Z}$, as $\rho(f) = (\lim_{n \to \infty} \tilde{f}^n(0)/n) + \mathbb{Z}...
- 616
7
votes
2
answers
780
views
Finite groups with a character having very few nonzero values?
A number theorist I know (who studies Galois representations) raised a question recently:
Which finite groups can have an irreducible character of degree at least 2 having only $n=2, 3$, or 4 ...
- 52.9k
6
votes
1
answer
2k
views
Generalization of the fundamental theorem of cyclic groups
Let $G$ be a finite group then the fundamental theorem of cyclic groups can be formulated as follows:
Theorem: $G$ is cyclic iff it admits no two different subgroups with the same order.
proof: see ...
6
votes
1
answer
587
views
What are the "simplest" polytopes with an automorphism group of $\mathrm M_{11} \hspace {-1.25pt} $?
Do any polytopes have an automorphism group of the smallest of the sporadic groups, the Matthieu group $\mathrm M_{11} \hspace {-1pt} $? Indeed, they must exist. What are the simplest such polytopes ...
- 179
6
votes
3
answers
2k
views
What is the motivation and purpose of the Floretion group?
When searching through the Oeis, I came across something called a floretion. Based on the context, it seems to be some sort of algebraic structure. I googled it and found nothing that explained their ...
- 1,129
6
votes
3
answers
348
views
Is there a maximal subgroup of depth 3?
Let's first define what we mean by depth of a subgroup.
Let $G$ be a finite group and $H$ a subgroup. Let $(V_i)_{i \in I}$ and $(W_j)_{j \in J}$ be the irreducible complex representations of $G$ ...
6
votes
2
answers
4k
views
Example of an infinite abelian but non-cyclic group whose automorphism group is cyclic
Can anyone give me an example of:
An infinite abelian but non-cyclic group whose automorphism group is cyclic.
- 4,795
6
votes
1
answer
162
views
Enlarging a subdegree-finite "almost transitive" permutation group to a transitive one? (follow-up)
Consider a permutation group $G$ acting on an infinite set $X$. Assume $G$ has finitely many orbits, and every point stabiliser $G_x$ has finite orbits.
Can we always find a permutation $\tau\in\...
- 327
6
votes
2
answers
693
views
Groups whose centralisers are finite
Let $G$ be an infinite group such that the centraliser of any non central element is finite (and bounded).
Is there any structure theorem known about $G$ ?
Such a group seems to be at the other ...
- 1,555
6
votes
1
answer
338
views
Semisimple compact Lie group topologically generated by two finite order elements
Edit: I'm specializing this question to the compact case I'll ask about the noncompact case as a new question.
Let $ G $ be a compact connected semisimple Lie group.
Do there always exist two finite ...
- 4,081
6
votes
0
answers
190
views
The highest degree of a polynomial on a finite group
This question is motivated by the comments and the answer to this MO-question.
First let us recall some definitions.
A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\...
- 41.9k
6
votes
2
answers
353
views
The Tits classes of simply connected simple real groups
Let $G$ be a simply connected semisimple group over a perfect field $k$ (at the moment I am interested in the case $k=\mathbb R$).
Then $G$ is an inner form of a quasi-split $k$-group $G_{\rm qs}$:
...
- 14.2k
6
votes
1
answer
828
views
Which functions are linear combinations of irreducible characters for a given field $\Bbbk$?
Let $G$ be a finite group. Then it is well known that a function $f\colon G\to \mathbb C$ is a linear combination of irreducible characters iff it is constant on conjugacy classes. What is the ...
- 38.6k
6
votes
1
answer
371
views
Embedding a cancellative monoid into another in such a way that a prescribed element becomes left-invertible
Let $\mathbb A = (A, +_A)$ be a cancellative, but possibly non-commutative, monoid with identity $0$, and fix an element $x \in A$. Does there always exist a cancellative monoid $\mathbb B = (B, +)$ ...
- 10.5k
6
votes
2
answers
575
views
Which groups are doubling?
A metric space $(M,d)$ is doubling if there exists $n$ such that every ball of radius $r$ can be covered by $n$ balls of radius $r/2$, for all $r$. For which f.g. groups $G$ and finite symmetric ...
- 6,652
6
votes
0
answers
453
views
Does the Approximation Property (AP) pass to quotients by amenable subgroups?
Given a countable group $G$ with the AP, and an normal, amenable subgroup $N$ of $G$, does $G/N$ have the AP?
In particular, does there exist a group $G$ with the AP and a surjective group ...
- 3,497
6
votes
0
answers
618
views
Duality between conjugacy classes and irreducible characters for finite monoids?
Qiaochu's answer to this question suggests that the proper way to view the bijection between conjugacy classes and irreducible complex representations of a finite group is via a duality. My question ...
- 38.6k
6
votes
2
answers
326
views
Is there a combinatorial interpretation of this array in terms of $S_{2n+1}$?
I have recently encountered a triangular array $(a_{i,j})_{0\le i\le j}$, each line of which might (should?) have a combinatorial interpretation in terms of $S_{2n+1}$. Here it is (the first entry of ...
- 13.4k
6
votes
1
answer
567
views
Finite simple groups and $ \operatorname{SU}_n $
A follow-up question to Alternating subgroups of $\mathrm{SU}_n $.
$\DeclareMathOperator\PU{PU}\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\PSL{PSL}$Let $ \PU_m $ be the projective unitary group, ...
- 4,081
6
votes
1
answer
845
views
Centraliser of the complex conjugation in the absolute Galois group
Is it known what is the centralizer of the complex conjugation in the absolute Galois group (i.e. the Galois group of the field of complex algebraic numbers over the rationals)? and, what would be ...
- 63
6
votes
2
answers
451
views
Stabilizer in automorphism group of free group of a certain finite-index subgroup
The following question arose in my research. I'd be interested in an answer to it, but I'd also be interested in general techniques for solving this kind of problem (or, even better, pointers to ...
- 207
6
votes
1
answer
317
views
Do one-relator groups satisfy Haagerup property?
The question is in the title:
Do one-relator groups satisfy Haagerup property?
I think the answer is known at least in some specific cases, but is the problem completely solved?
- 2,371
6
votes
1
answer
431
views
Do free profinite groups satisfy Howson's theorem?
Let $F$ be a free profinite group, and let $A,B \leq F$ be finitely generated closed subgroups. Must $A \cap B$ be finitely generated?
- 11.3k
6
votes
1
answer
849
views
amalgamated product of groups and representation theory
Let me ask a question which could be quite stupid, but still:
let $G$ be a group which is an amalgamated product of subgroups $A$ and $B$ over $C$:$\; \;$ $G = A \ast_{C} B$ (subgroups are infinite!)...
- 143
6
votes
0
answers
245
views
Wild automorphisms of a free group
Let $F_X$ be a free group on a countably infinite set $X$. Let $\alpha$ be an automorphism of $F_X$ and $H$ a closed subgroup of $F_X$ in the profinite topology.
Is it possible that $\alpha(H) \...
- 11.3k
6
votes
4
answers
889
views
On the determination of a quadratic form from its isotropy group
Let $F:\mathbf{R}^n\rightarrow\mathbf{R}$ be a non-degenerate quadratic forms. Let
$$
O(F):=\{g\in GL_n(\mathbf{R}):F(gv)=F(v),\forall v\in \mathbf{R}^n\}
$$
be the isotropy group of $F$.
Q: So how ...
- 7,609
6
votes
1
answer
406
views
Connection between Stalling's end theorem and Seifert-van Kampen Theorem
Stalling`s end Theorem (a group has more than one end iff it splits over a finite subgroup) and the Seifert-van Kampen Theorem (the fundamental group of a 'decomposable' space is a free amalgamated ...
- 721
6
votes
3
answers
2k
views
Do the homological dimension and cohomological dimension for a group agree?
Or equivalently, if $G$ is a group, do the projective and injective dimension of $Z$ (viewed as a $ZG$-module) agree?
Thanks!
- 113
6
votes
1
answer
338
views
How large a subset do you need to uniquely determine a 2-cocycle?
Suppose A and B are abelian groups. I want to find subsets D of $A \times A$ such that any 2-cocycle $c:A \times A \to B$ for the trivial action is uniquely determined by what it does on D. (...
- 7,320
6
votes
2
answers
711
views
maximal tensor product of simple $C^*$algebras is non-simple
Let $A$ and $B$ simple $C^*$-algebras. One can prove that the minimal tensor product $A\otimes _{min}B$ is simple. This is wrong for the maximal tensor product $A\otimes_{max}B$ .
1.Do you know an ...
- 1,188
6
votes
5
answers
3k
views
Group cannot be the union of conjugates
I have seen that if $G$ is a finite group and $H$ is a proper subgroup of $G$ with finite index then $ G \neq \bigcup\limits_{g \in G} gHg^{-1}$. Does this remain true for the infinite case also?
- 4,795
6
votes
2
answers
694
views
Can the image of a Schur functor always be made an irreducible representation?
For a partition $\lambda$ let $S^{\lambda}$ be the corresponding Schur functor. Is it true that for every $\lambda$ there exists an irreducible representation $V$ of a finite nonabelian group $G$ ...
- 118k
6
votes
1
answer
444
views
Relations between relations in the positive braid monoid
The positive braid monoid on $n$ strands is the monoid with generators $s_1$, $s_2$, ..., $s_{n-1}$ and relations
$$s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1} \qquad s_i s_j = s_j s_i \text{for}\ |i-j| \...
- 156k
6
votes
0
answers
172
views
Finite subquotients of R. Thompson's group $F$
Recall R. Thompson's group $F$ acting on the interval $[0,1]$: it consists of piecewise linear oriented maps with slopes a power of $2$ and dyadic breakpoints.
Is every finite subquotient (= quotient ...
- 2,519
6
votes
0
answers
430
views
More on Moonshine for the Thompson group and weakly holomorphic weight one half modular forms
This question is a follow-up to
Monstrous Moonshine for Thompson group $Th$?
and is based on various comments to that question, in particular S. Carnahan's mention of the
connection to known ...
- 5,546
6
votes
2
answers
2k
views
If $G \times G \cong H \times H$, then is $G \cong H$? [duplicate]
Let $G,H$ groups. Suppose that $G \times G \cong H \times H$. Then is necessarily $G \cong H$?
I know that if $G,H$ are finite groups then it is true (you show that the map $FinGrp \rightarrow \...
- 69
6
votes
1
answer
355
views
Do doubly-transitive actions give rise to indecomposable representations for infinite groups?
This is a follow-up to this question.
Let $G$ be a group acting doubly-transitively on a set $X$. Then the vector space $V_X$ of functions $f\colon X\to\mathbb C$ with finite support such that $\sum_{...
- 3,054
6
votes
1
answer
226
views
Are $G$-limits of a slender group $G$ in the space of marked groups also slender?
A group $G$ is slender if every subgroup $H \leq G$ is finitely generated. This includes polycyclic-by-finite groups. Such groups are also called noetherian.
Suppose that $L$ is a $G$-limit group in ...
- 1,033
6
votes
1
answer
678
views
Finite Homomorphic images of infinite products of finite solvable groups
I conjecture that:
Every Finite Homomorphic image of an infinite (with arbitrary cardinality) product of finite solvable groups is solvable -- or at least Not a simple (non-abelian) group.
I can ...
- 145
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
492
views
Centralizer of elements in the upper-triangular matrices
Let $p$ be a prime number and $G=\operatorname{GL}_n ( \mathbb{Z} / p \mathbb{Z}
)$ such that $n\leq p$. Consider the set $U$ of upper-triangular matrices of $G$
having entries of $1$ on the diagonal. ...
- 463
6
votes
1
answer
2k
views
Are there infinitely many insipid numbers?
A number $n$ is called insipid if the groups having a core-free maximal subgroup of index $n$ are exactly $A_n$ and $S_n$. There is an OEIS enter for these numbers: A102842. There are exactly $486$ ...
6
votes
1
answer
435
views
Dynkin diagram of the centralizer of a semisimple element in a Levi subgroup
Let $G$ be a connected reductive group over an algebraically closed field and consider a semisimple element $s \in G$ and let $L$ be a Levi subgroup containing $s$.
My question is about the two ways ...
- 309