Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
1,095 questions
9
votes
4
answers
1k
views
Elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$
For some research work, I need to know the classification of elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$, up to conjugation.
Since I essentially need $n\le 4$, I think that I can show it ...
- 7,722
9
votes
1
answer
384
views
Reference for the fact that $SL_n(O_K)$ surjects onto $SL_n(O_K/I)$ for any ideal I
Let $\mathcal{O}_K$ be the ring of integers in an algebraic number field $K$ and let $I \subset \mathcal{O}_K$ be a nonzero proper ideal. It is not hard to see that the map $\text{SL}_n(\mathcal{O}_K)...
- 91
9
votes
2
answers
1k
views
When is the profinite completion a pro-$p$ group?
My research area is mainly pro-$p$ groups and profinite groups. However, in the last few year I became also interested in discrete groups. Therefore, it seems to me a natural problem to look for ...
- 5,450
9
votes
1
answer
394
views
Is every automorphism of $\mathrm{Aut}^+(F_2)$ induced by conjugation inside $\mathrm{Aut}(F_2)$?
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $F_2$ be a free group of rank 2. There is a surjection $\Aut(F_2)\rightarrow \GL(2,...
- 7,527
9
votes
2
answers
2k
views
Which finite groups are generated by n involutions?
One of the interesting problems in abstract polytope theory is to determine, for a given finite group, when that group is the automorphism group of a regular abstract polytope. This is equivalent to ...
- 1,861
9
votes
0
answers
445
views
Which finite solvable groups have solvable automorphism groups?
Is it possible to give a reasonable description of those finite solvable groups $G$ such that $A = {\rm Aut}(G)$ is also solvable?
The central case to deal with is that in which $G$ is a $p$-group of ...
- 44.4k
9
votes
1
answer
582
views
Do doubly-transitive actions give rise to irreducible representations for infinite groups?
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 $\sum_{x\in X}f(x)=0$ carries an action of $G$...
- 3,054
9
votes
1
answer
521
views
Is a free group a product of f.g subgroups of infinite index?
Let $F$ be a free group, and let $H,K \leq F$ be finitely generated subgroups of infinite index in $F$. Is it possible that for the set of products we have $HK = F$ ?
- 11.3k
9
votes
1
answer
356
views
Diameter of the modified bubble-sort graph
The modified bubble-sort graph is the Cayley graph $Cay(S_n,S)$ of $S_n$ generated by $n$ cyclically adjacent transpositions. Thus $S = \{ (1,2),(2,3),\ldots,(n,1)\}$. I was wondering whether the ...
- 406
9
votes
3
answers
842
views
Is there a one relator group with property (T)?
Is there a one-relator group with property (T)?
That is, is there an $n > 2$, and some $x \in F_n$ (the free group on $n$ generators) such that the quotient of $F_n$ by the normal subgroup ...
- 11.3k
9
votes
3
answers
947
views
Where was it first stated that there are no 4-transitive finite groups other than symmetric, alternating and Mathieu groups?
It seems to be well-known that the six-transitive finite groups are the symmetric and alternating groups, and the only other four-transitive finite groups are the Mathieu groups (the statement can be ...
- 96.4k
9
votes
1
answer
1k
views
Fundamental group of an hyperbolic $4$-manifold
Good afternoon everyone,
I have a very general question about hyperbolic manifolds and their fundamental groups in high dimension (at least $4$). If the theory of surfaces and $3$-manifolds provide ...
- 2,696
9
votes
3
answers
1k
views
First-order axiomatization of free groups
Is there a way to axiomatize [non-abelian] free groups in first-order logic using the language of groups (which contains the binary operation symbol $\cdot$, and the constant symbol $e$)?
Is there ...
- 39.8k
8
votes
1
answer
351
views
Which reflection groups can be enlarged?
Based on this question (which focuses on the case $E_8$) I wonder the following:
Question: For each finite reflection group $\Gamma\subseteq\mathrm O(\Bbb R^d)$, what is the largest finite group $\...
- 13.6k
8
votes
1
answer
619
views
Amenable groups of finite cohomological dimension
I seek an example of an amenable group of finite cohomological dimension that is not virtually polycyclic and has finite abelianization.
Remarks.
Elementary amenable groups of finite cohomological ...
- 29.1k
8
votes
4
answers
2k
views
Bounds on number of conjugacy classes in terms of number of elements of a group ?
What are bounds on number of conjugacy classes in terms of number of elements of a group ?
(I allowed myself to edit the question in spirit of remarkable answers given to it by Gerry Myerson and ...
- 375
8
votes
2
answers
489
views
Quantitative word problem for 3-manifold groups
The word problem for 3-manifold groups is solvable: given a based loop $\gamma$ in $M^3$ there is an algorithm to decide whether $\gamma$ bounds a disk.
What kinds of quantitative results are known ...
- 595
8
votes
0
answers
304
views
A strong sum-product "for translates" in finite fields
In the course of some recent research, I've sketched out a proof of the following result. My basis question is: is the result interesting?
Proposition There exists an absolute constant $c$ such ...
- 11.2k
8
votes
1
answer
256
views
Existence of a finite group with a given decomposition for a tensor square of one irreducible complex representation
In this post, irrep and dim mean "irreducible complex representation" and "dimension", respectively.
It would be helpful (in a problem of monoidal category) to find a finite group $G$ with (at least) ...
8
votes
1
answer
572
views
Mostow Rigidity Theorem and reconstruction from fundamental group
The Mostow Rigidity Theorem is phrased in terms of a relationship between isometries and isomorphisms of fundamental groups, which raises an obvious question. Given the fundamental group of a complete ...
- 3,816
8
votes
1
answer
731
views
Normal subgroup of classical groups
For $r\leq n$, consider the following reduction homomorphism
$$
\pi_{n,r}: {\rm SL}_2(\mathbb{Z}/(p^n\mathbb{Z}))\to {\rm SL}_2(\mathbb{Z}/(p^r\mathbb{Z})).
$$
Bourgain and Gamburd in their paper "...
- 2,508
8
votes
2
answers
586
views
How hard is it to compute the diameter and the growth function of a finite permutation group of small degree?
Let $G \leq {\rm S}_n$ be a finite permutation group, and let
$S = \{g_1, \dots, g_k\}$ be a generating set for $G$ which is closed
under inversion and which does not contain the identity.
The growth ...
- 19.6k
8
votes
1
answer
948
views
Surjective maps given by words, redux
I asked some time ago:
Let $w(X,Y)$ be a word in $X$ and $Y$ (i.e., an element in the free group on $X$ and $Y$). Let the variables $x$ and $y$ now range among elements of $SL_n(K)$, $K$ an ...
- 20.2k
8
votes
1
answer
435
views
Minimal number of generators of subgroups of Noetherian groups
A group $G$ is Noetherian (or slender) if all its subgroups are finitely generated. Does this imply that the minimal number of generators of subgroups of $G$ is bounded above?
For example, if $G$ is ...
- 937
8
votes
2
answers
799
views
characters on a finite group with `extremal' behaviour
The following question is a bit technical, and I haven't got to grips with it enough to be able to present it as a well-focused question. However, my hope is that the collective group-theoretic ...
- 25.8k
8
votes
3
answers
559
views
Reference for tetrahedral Coxeter group
Let $G$ be the group with 4 generators, each of order 2, such that the product of any 2, say $ab$, has order 3 (i.e., $ababab=e$).
That is, this is an infinite reflection group with Coxeter diagram a ...
- 83
8
votes
2
answers
2k
views
Expectation of trace of nth power of unitary matrices
I am trying to find the answer of
$$\int dU \ |Tr(U^m)|^2$$
where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $\textit{U}(n)$ and $dU$ is a normalized Haar measure. In the case $m=1$, the ...
- 127
8
votes
0
answers
185
views
Sharp isoperimetry in the discrete Heisenberg group
The exact shape of the set which has the best isoperimetry in the continuous Heisenberg is (from what I know) a difficult open problem. This brought to wonder what is known in the discrete case?
More ...
- 4,432
8
votes
1
answer
1k
views
Finite groups containing no subgroups of a given order or index
The classical Lagrange's Theorem says that the order of any subgroup of a finite group divides the order of the group. For abelian groups this theorem can be completed by the following simple fact: ...
- 41.9k
8
votes
1
answer
495
views
What is the Schur multiplier of the affine linear group AGL(n,q)?
What is the Schur multiplier of the $n$-dimensional affine linear group $\mathrm{AGL}(n,q)$ over the Galois field with $q$ elements?
I am particularly interested in the simple case $n=1$. Computation ...
- 533
8
votes
1
answer
896
views
Quantifier elimination for abelian groups
In the Wikipedia article (https://en.wikipedia.org/wiki/Quantifier_elimination#cite_note-4) it is said that every abelian group has quantifier elimination property and a long old paper of W. Szmielew ...
- 2,233
8
votes
0
answers
545
views
What are the character tables of the finite unitary groups?
I need to know the (complex) character table of the finite unitary group $U_n(q)$. Lusztig and Srinivasan (1977) provide an abstract description, but parsing it requires a stronger background in ...
- 7,633
8
votes
2
answers
853
views
Is a Hopf algebra a group object of some category?
The page of ncatlab on group object states that:
A group object in $\mathrm{CRing}^{\mathrm{op}}$ is a commutative Hopf
algebra.
Question: Is a (noncommutative) Hopf algebra a group object of some ...
8
votes
1
answer
362
views
If a group $G$ has decidable word problem, must it have a decidable square problem?
My question is a refinement of this one about 'efficient' construction of square elements: If the word problem for a (finitely generated, finitely presented) group is decidable, must the 'square ...
- 1,951
8
votes
2
answers
750
views
Avoiding countable subgroups of general uncountable groups
The following problem is a general form of another problem (motivation is available there). Initially, the problems were posted together, but the first one is solved below, a solution that does not ...
- 3,104
8
votes
1
answer
274
views
On the number $n_0$ in Shelah's construction of a Jonsson group
In the paper "On a Problem of Kurosh, Jonsson Groups, and Applications", Shelah proved the following
Theorem 2.1. There exists a number $n_0$ such that for every infinite cardinal $\lambda$ with $\...
- 41.9k
8
votes
5
answers
1k
views
Subgroups of a group generated by a free semigroup
Let $F$ be a free semigroup (say, $2$-generated) which is embedded in a group $G$, and suppose that $G$ (as a group) is generated by $F$. The most simple such situation would be when $G$ is a free ...
- 2,804
8
votes
0
answers
203
views
Word maps from $\mathrm{Cl}^{n+1}$ to $G^n$: quasi-injectivity?
Let $G = \mathrm{SL}_n$ (say). Then, for $g$ regular semisimple, the conjugacy class $\mathrm{Cl}(g)$ has dimension $= n^2-n$ as a variety, and so $\mathrm{Cl}(g)^{n+1}$ and $G^n$ have the same ...
- 20.2k
8
votes
1
answer
672
views
Can the Kan-Thurston theorem be turned into some kind of equivalence between groups and spaces?
I not really familiar with these subjects. I read this question and I was really surprised by the answer. My question is probably vague (so please do bear with me).
The cited question/answer ...
- 277
8
votes
2
answers
1k
views
What does the unique mean on weakly almost periodic functions look like?
There is a unique invariant mean $m$ on WAP functions on any discrete group (see definitions below, theorem of ?). However, the proofs I found are fairly non-explicit on how to obtain this invariant ...
- 4,432
8
votes
1
answer
360
views
A combinatorial property of uncountable groups, II
Problem 1. Is it true that each uncountable group $G$ contains two subsets $A,B\subset G$ such that
1) for any $x,y\in G$ the intersection $xA\cap yB$ is finite and
2) for any function $\...
- 41.9k
8
votes
2
answers
367
views
Existence of a special ordering of the elements of a finite group
Let $G$ be a finite non-abelian group of order $n$. Given $g\in G$ we denote its order by $\mathrm{ord}(g)$.
Consider the group algebra $\mathbb{F}[G]$ for some field $\mathbb{F}$.
Given an ordering $...
- 14.6k
8
votes
1
answer
562
views
The parity of the full automorphism group order of finite non-abelian groups of prime exponent
Is there a finite non-abelian group $G$ of prime exponent such that the full automorphism group of $G$ is of odd order?
- 3,738
8
votes
1
answer
740
views
Is $Spin(N)$ a subgroup of $SU(N)$
Is $Spin(N)$ a subgroup of $SU(N)$? If so, how can we embed $Spin(N)$ into $SU(N)$? I would love to find a representation where both $Spin(N)$ and $SU(N)$ act faithfully and see explicitly how the ...
- 169
8
votes
3
answers
2k
views
Are congruence subgroups of the modular group finitely presented?
Are the congruence subgroups of the modular group $\Gamma\equiv\mathrm{PSL}\left(2,\mathbb{Z}\right)$ (e.g. $\Gamma\left(n\right)$, $\Gamma_{0}\left(n\right)$, $\Gamma_{1}\left(n\right)$ etc.) ...
- 393
8
votes
2
answers
600
views
How many minimum generating sets are there in a finite group?
Let $G$ be a finite group of order $n$.
A generating set in $G$ is said to be minimum if it has minimal size.
Is there a known lower bound on number of minimum generating sets in a group of order $...
user108347
8
votes
2
answers
404
views
Homomorphisms from $\mathbb{R}$ to $\mathrm{Homeo}^+(\mathbb{R})$, or "fractional iterations"
Let $G$ be the group of orientation-preserving homeomorphisms (or, if you prefer, diffeomorphisms) of the real line. Does there exist a natural way to associate, to each function $f \in G$, a ...
- 7,338
8
votes
0
answers
666
views
Approximating Lie groups by finite groups
How can one approximate compact Lie groups by finite groups?
My wish is something like this:
Let $G$ be a compact Lie group.
There is a sequence of nested finite subgroups $G_n$ so that $G_n\to G$...
- 8,086
8
votes
1
answer
153
views
Are there Type III codes with small but nonzero "index"?
Recall that a Type III code of rank $r$ is a linear subspace $C \subset \mathbb F_3^r$ which is self-dual for the standard inner product. (These occur only when $r$ is divisible by $4$.) Elements of $...
- 54.6k
8
votes
4
answers
659
views
Normal Covering of a Finite Group
Suppose $G$ is a finite group and $N_1, N_2, \cdots, N_k$ are proper normal subgroups of $G$. The set $\{ N_1, \cdots, N_k\}$ is called a normal cover for $G$, if $G = \cup_{i=1}^kN_i$. I need to the ...
