Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,182 questions
15
votes
0
answers
477
views
Diffeomorphisms of $\mathbf R^n$
Let $G={\rm Diff}_0^c(\mathbf R^n)$, $n\geq 1$, be the group of compactly supported diffeomorphisms isotopic to the identity through compactly supported isotopies.
Question: Is there an example to ...
- 1,782
15
votes
0
answers
189
views
Quantitative form of Wielandt's theorem
The following theorem was proved in [Helmut Wielandt. Eine Verallgemeinerung der invarianten Untergruppen. Mathematische Zeitschrift 45 (1939): 209-244.] a long time ago:
Theorem: (Wielandt) There ...
- 25.5k
15
votes
0
answers
716
views
Is this "Homology" useful to study?
In the usual singular homology of a topological space $X$, one consider the free abelian group generated by all continuous maps from the standard simplex $\Delta^{n}$ to $X$.
Now we can ...
- 356
15
votes
0
answers
573
views
Relation Between Truncated Braid Groups and Regular Tilings of the Complex and Hyperbolic Plane
This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question on math.SE.
There exists a rather remarkable ...
- 715
15
votes
0
answers
745
views
Galois theory: Generalization of Abel's Theorem?
Let $L$ stand for the field obtained by adjoining to
${\Bbb Q}$ all roots of all polynomials of the
form $x^n+ax+b$, $a,b\in {\Bbb Q}$.
What polynomials $p$ don't split over $L$? In particular, ...
- 17.6k
15
votes
0
answers
753
views
Are all these groups CAT(0) groups?
Given a geodesic metric space $X$ together with a choice of midpoints
$m:X\times X\rightarrow X$ (i.e. $d(m(x,y),x)=d(m(x,y),y)=d(x,y)/2$).
Assume furthermore, that the following nonpositive curvature ...
- 11.1k
14
votes
5
answers
663
views
Results about the order of a group forcing a particular property.
Given a group of order $n$ where $n$ is either a specific number, or a number of a particular form, e.g. square-free, when does $n$ completely determine a particular group property among all groups of ...
- 483
14
votes
2
answers
1k
views
$n!$ divides a product: Part I
Question. The following is always an integer. Is it not?
$$\frac{(2^n-1)(2^n-2)(2^n-4)(2^n-8)\cdots(2^n-2^{n-1})}{n!}.$$
John Shareshian has supplied a cute proof. I'm encouraged to ask:
...
- 43.2k
14
votes
8
answers
2k
views
less elementary group theory
Most of the group theory that is taught in introductory graduate classes is of the form $$(\mbox{number theory} + \mbox{ group actions} + \mbox{ orbit-stabilizer thm}) + \mbox{group axioms} \...
14
votes
3
answers
1k
views
Countable subgroups of compact groups
What is known about countable subgroups of compact groups? More precisely, what countable groups can be embedded into compact groups (I mean just an injective homomorphism, I don't consider any ...
- 1,318
14
votes
3
answers
780
views
Deciding if $\mathbb{Z}\ltimes_A \mathbb{Z}^5$ and $\mathbb{Z}\ltimes_B \mathbb{Z}^5$ are isomorphic or not
I asked this in this MSE question but I didn't get answers. I think maybe here someone can help me.
I have the two following groups
$G_A=\mathbb{Z}\ltimes_A \mathbb{Z}^5$, where $A=\begin{pmatrix} 1&...
14
votes
4
answers
697
views
Non-split Aut(G) $\to$ Out(G)?
I'm looking for examples of outer automorphisms of a finite group $G$ which do not lift to automorphisms (i.e. non-split quotient map $\mathrm{Aut}(G)\to \mathrm{Out}(G)$, where $\mathrm{Out}(G) = \...
- 12.8k
14
votes
5
answers
5k
views
Finite groups with elements of order n
Consider a finite group where all elements have the same order $n$.
What could be said about such groups?
For $n=2$ it could be proved that such group is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^k$.
...
- 2,821
14
votes
2
answers
906
views
Acyclic group and finite CW-complex
Is there a nontrivial example of an acyclic group $G$ such that its corresponding Eilenberg space $K(G,1)$ is homotopy equivalent to a finite CW-complex ?
- 717
14
votes
4
answers
2k
views
Number of squares in a finite group
This was asked at MSE but never answered.
Let $G$ be a finite group and denote by $sq(G)$ the number of squares in $G$ i.e. the number of elements in $G$ which possess a square root. For example, if
...
- 1,411
14
votes
3
answers
2k
views
Minimal generating sets of groups
I am not exactly a group theorist, so this may be well-known.
Let $G$ be a finitely generated group such that the cardinality of minimal generating sets of $G$ is bounded above.
Does it follow that $...
- 1,072
14
votes
7
answers
3k
views
Cheap, non-constructive, free group generating rotations for Banach-Tarski
Stan Wagon's exposition of Banach-Tarski (for example) includes a beautiful explicit construction of two 2-sphere rotations which generate a free subgroup of the rotation group.
For teaching purposes ...
- 17.6k
14
votes
1
answer
1k
views
A nice problem by Peter Cameron on subsets of $\{1,\dots,n\}$
Recently Professor Peter Cameron posed a number theory problem which is related to graphs of groups. The problem is:
Problem:
Let $n$ be a positive integer. Show that there exist subsets $A_1, A_2, …,...
- 4,784
14
votes
2
answers
1k
views
On the finite simple groups with an irreducible complex representation of a given dimension
This answer of Geoff Robinson shows that a finite simple group admits an irreducible complex representation (irrep) of dimension $3$ if and only if it is isomorphic to $A_5$ or $\mathrm{PSL}(2,7)$.
...
14
votes
2
answers
922
views
Units in group rings.
Let $G$ be a finite solvable group of order $n$, and let $g_1 ... g_n$ be an enumeration of its elements. Let $a_1 ... a_n$ be a sequence of integers, such that $\sum a_i$ is relatively prime to $n$.
...
- 191
14
votes
3
answers
1k
views
Is there any criteria for whether the automorphism group of G is homomorphic to G itself?
In the elementary group theory we know that for the symmetric groups $S_n$, except $S_6$, we have $Aut(S_n) \cong S_n$. Then the following question is natural:
What is the necessary and sufficient ...
- 627
14
votes
3
answers
683
views
Compact manifolds with big mapping class group
I was wondering if compact surfaces were the only compact manifolds with a "big" or "complicated" mapping class group.
Are there higher dimensional manifolds (which are not in some way reducible to ...
- 2,696
14
votes
4
answers
3k
views
When does Pontryagin duality generalize?
Let $T$ be a locally compact abelian (LCA) group. For any other LCA group $G$, let
$\hom(G,T)$ be the set of continuous homomorphisms $G\to T$. With the compact-open
topology, $\hom(G,T)$ is ...
- 5,759
14
votes
5
answers
2k
views
A general formula for the number of conjugacy classes of $\mathbb{S}_n \times \mathbb{S}_n$ acted on by $ \mathbb{S}_n$
$\def\S{\mathbb{S}}$ Dear all,
So I have $\S_n$ acting on $\S_n \times \S_n$ via conjugacy. That is:
for $g \in \S_n, (x,y) \in \S_n \times \S_n$: $g(x,y) = (gxg^{-1},gyg^{-1}).$
Is there a general ...
- 1,010
14
votes
3
answers
3k
views
Presentation of the Monster Group
I was reading about the monster group, and how hard it was to do calculations in it, and I wondered: Is there a known presentation of the monster group? I know that it is a hurwitz group, but other ...
- 2,811
14
votes
4
answers
778
views
Largest permutation group without 2-cycles or 3-cycles
The largest permutation group without 2-cycles is $A_n$, which has size $n!/2$. I think the largest permutation group without 2-cycles or 3-cycles is much smaller, but I can't figure out if it should ...
- 143
14
votes
2
answers
852
views
Examples of finitely presented subgroups of $\operatorname{GL}(n,\mathbb{Z})$ with unsolvable decision problems
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Does there exist a finitely presented subgroup of $\GL(n,\mathbb{Z})$ for which it is known that the conjugacy problem is unsolvable (if yes, ...
- 143
14
votes
2
answers
1k
views
If every definable class admits a group structure, must global choice hold?
It is a remarkable fact, due to Hajnal and Kertész and explained very well in this MathOverflow answer by user Ashutosh, that the axiom of choice is equivalent to the assertion that every nonempty set ...
- 236k
14
votes
3
answers
660
views
Which partitions realise group algebras of finite groups?
Fix an algebraically closed field $K$ (maybe of characteristic zero first for simplicity, like $\mathbb{C}$).
Given a partition $p=[a_1,...,a_m]$ of an integer $n$. We can identify $p$ with the ...
- 26.5k
14
votes
2
answers
725
views
Are there any non-conjugation "extendible automorphisms" in the category of finite groups?
Let $\mathbf{Grp}$ be the category of groups. Given a subcategory $\mathscr{G}$ of $\mathbf{Grp}$ and $G\in\mathit{Ob}(\mathscr{G})$, a $\mathscr{G}$-extendible map on $G$ will here mean an assignment ...
- 20.5k
14
votes
2
answers
785
views
Semisimple representations of discrete groups
Let $G$ be a discrete group. Let $V$ and $W$ be two finite dimensional complex simple $G$ representations.
QUESTION. Must the tensor product $V\otimes_{\mathbb{C}} W$ with the diagonal action be a ...
- 5,039
14
votes
2
answers
1k
views
Can a group generated by its involutions, the product of every two of which has order a power of 2, have an element of odd order?
Let $G$ be a group which is generated by the set of its involutions,
and assume that the product of every two involutions in $G$ has order
a power of 2. Is it possible that $G$ has an element of odd ...
- 19.6k
14
votes
3
answers
1k
views
Does the Alternating group of degree $n>7$ have exactly one irreducible character of degree $n-1$?
We know that the alternating group of degree $n>7$ has an irreducible character of degree $n-1$. The latter number is the smallest nontrivial one for each the alternating group has an irreducible ...
- 3,738
14
votes
2
answers
985
views
Has the Jacobson/ Baer radical of a group been studied?
On groupprops, the Jacobson or Baer radical of a group $G$ is defined to be the intersection of all maximal normal subgroups of $G$. This is similar to, but distinct from, the Frattini subgroup which ...
14
votes
2
answers
863
views
Example of a ring with non-finitely generated unit group?
The well known Dirichlet's unit theorem states that the unit group of a maximal order in a quadratic number field is finitely generated of rank blah blah blah. I think it's pretty naive to expect a ...
- 4,600
14
votes
2
answers
1k
views
When are growth series rational?
For a finitely generated group $G$ with generating set $S$, one can define the word metric. Using this, we are able to define the sequence $\sigma(n)$ by
$ \sigma(n) = |\mathcal{S}(e,n)| $, which is ...
14
votes
2
answers
634
views
Non-congruence normal subgroups of $SL_2(\mathbb{Z}[1/2])$
Let $G=SL_2(\mathbb{Z}[1/2])$, i.e., the modular group (if you wish) over the ring $\mathbb{Z}[1/2]$ consisting of rationals whose denominators are powers of $2$. Unlike $SL_2(\mathbb{R})$, $G$ is ...
- 20.2k
14
votes
4
answers
631
views
Normal subgroups of an extension of the Higman group
Let $G = \mathbb{Z}/4\mathbb{Z} \ltimes H_4$, where $H_4$ is the Higman group and $\mathbb{Z}/4\mathbb{Z}$ acts on $H_4$ in the obvious way (permuting the four standard generators cyclicly). The group ...
- 20.2k
14
votes
2
answers
748
views
Checking whether given binary operation is a group operation
Given a binary function $f: [1..n] \times [1..n] \to [1..n]$ how to check that this operation is a group operation on $[1..n]$?
It's obvious that this can be done in $O(n^3)$ time just by checking ...
- 2,821
14
votes
2
answers
1k
views
The maximum order of finite subgroups in $GL(n,Q)$
For several years people have tried to characterize finite groups of maximal order in $\mathrm{GL}(n,\mathbb{Q})$ (or $\mathrm{GL}(n,\mathbb{Z})$) and their orders.
It appear in many articles a ...
- 2,829
14
votes
2
answers
980
views
Explicit abelianization functor for groups
Assume that I have a short exact sequence of finitely presented groups $$1 \longrightarrow K \longrightarrow H \longrightarrow G \longrightarrow 1,$$ where $G$ is finite (but I do not know whether ...
- 66.3k
14
votes
2
answers
873
views
Does every group grow either polynomially or superpolynomially?
I am reading an introduction to growth of groups. The notions of polynomial and superpolynomial growth are introduced, as are exponential and subexponential growth.
I can prove that the growth of a ...
- 655
14
votes
3
answers
488
views
Is every finitely-presentable group a finite colimit of copies of $F_2$?
Let $F_n$ be the free group on $n$ generators. Of course, every finitely-presentable group $G$ is a finite colimit of copies of $F_n$, where $n$ is allowed to vary. But is $G$ a finite colimit of ...
- 63.9k
14
votes
2
answers
1k
views
Groups acting on trees
Assume that $X$ is a tree such that every vertex has infinite degree, and a discrete group $G$ acts on this tree properly (with finite stabilizers) and transitively.
Is it true that $G$ contains a ...
- 631
14
votes
2
answers
2k
views
Explicit cocycle for the central extension of the algebraic loop group G(C((t)))
Let $G$ be a simple Lie group and let $G(\mathbb{C}((t)))$ be its loop group.
The Lie algebra $\mathfrak{g}[[t]][t^{-1}]$ has a well known central extension
(see e.g.
Wikipedia) given by the cocycle
...
- 43.2k
14
votes
1
answer
498
views
Abelianization of $\mathrm{GL}_n(\mathbb{Z})$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$What is the abelianization of $\GL_n(\mathbb{Z})$? I know the abelianzation of $\GL_n(\mathbb{F})$ where $\mathbb{F}$ is a field and the ...
- 911
14
votes
2
answers
1k
views
Quasi-isometry groups of metric spaces
Given a metric space $(X, d)$, we can consider the set of all quasi-isometries $f: X \to X$, and quotient out by the equivalence relation identifying $f$ and $g$ if $\sup_{x \in X}d(f(x), g(x))$ is ...
- 495
14
votes
2
answers
416
views
Schur multiplier of $Sp(2g, \mathbb{Z}/2)$ for $g \geq 3$
This question is about the computation of $H_2(Sp(2g, \mathbb{Z}/2), \mathbb{Z})$, where $Sp(2g, \mathbb{Z}/2)$ is the group of symplectic $2g \times 2g$ matrices over $\mathbb{Z}/2$.
With respect to ...
- 183
14
votes
1
answer
790
views
Order of elements
Consider natural numbers $m,n,k > 1$. There are finite groups $G$ containing elements $x,y$ such that $o(x) = m, o(y) = n$ and $o(xy) = k$. After embedding these groups in $S_\mathbb{N}$ we drive: ...
user30378
14
votes
2
answers
1k
views
Random walks on Coxeter groups
Let $G_N$ be the group generated by elements $a_1,\ldots,a_N$ subject to the relations $a_i^2=1$ and $(a_ia_j)^3=1$. The growth function of $G_N$ is then
$$f_N(t)=\frac{1+2t+2t^2+t^3}{1-Mt-Mt^2+\frac{...
- 1,363