Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,182 questions
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Is this class of groups already in the literature or specified by standard conditions?
In recent work
Lifting $N_\infty$ operads from conjugacy data on homotopical combinatorics / $N_\infty$ operads in equivariant homotopy theory, collaborators
Scott Balchin, Ethan MacBrough, and I ...
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Surprisingly only real points on intersection of certains quadrics
Let $G$ be a finite group and let $X_g$ be variables indexed by $G$. Consider the complex algebraic set defined by
\begin{align}
X_e &= 0\\
X_g &= X_{g^{-1}}\;\;\text{ for all }g\in G,\\
X_g &...
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Which irreducible representations of the symmetric group are eigenspaces of class sums?
In the setting of complex representations of finite groups, a class sum $1_C=\sum_{g\in C} g$ acts on an irreducible representation $V$ as $\lambda(C,V)\operatorname{Id}$, where $\lambda(C,V)=|C|\...
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Open questions about automorphism tower theorem
Joel Hamkins has left four open questions about automorphism tower theorem in his wonderful paper Every group has a terminating transfinite automorphism tower.
In fact, the four questions are "...
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Algebra for the Baby
I am reading the following article.
Ryba, Alexander J.E., A natural invariant algebra for the Baby Monster group., J. Group Theory 10, No. 1, 55-69 (2007). ZBL1228.20012..
Author works with 4370-...
user21230
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What is the mathematical name for the anomaly for an action of a group on a lattice conformal field theory?
Suppose $V$ is a (bosonic) chiral conformal field theory which is "holomorphic" in the sense that its category of vertex modules is trivial. (The definition of "chiral conformal field theory" might be ...
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Finite quotients of amalgamated products with virtually nilpotent factors
Consider the amalgamated product $A\ast_C B$ of groups such that $A\neq C\neq B$ and both factors $A$, $B$ are finitely generated virtually nilpotent.
Does $A\ast_C B$ always have a subgroup of some ...
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Does the category of G-spectra know G?
I was recently in the situation of having access to the category of $G$-modules (for some group $G$ which I had forgotten), as just a category, i.e. no monoidal structure, together with the forgetful ...
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Is $\mathbb{Z}^{\omega}$ ever the union of a chain of proper subgroups each isomorphic to $\mathbb{Z}^{\omega}$?
Recall that the covering number $cov(B)$ is the least cardinal $\kappa$ such that $\kappa$ meagre sets cover the real line. Andreas Blass and John Irwin http://www.math.lsa.umich.edu/~ablass/bb.pdf ...
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Is there an infinite field of characteristic 2 whose multiplicative group is torsion free and (direct-sum) indecomposable?
Let $F$ be a infinite field of characteristic 2 whose multiplicative group $F^*$ is torsion free. I would like to conclude that $F^*$ is decomposable or find an example where $F^*$ is indecomposable.
...
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Are there additive subgroups of reals of dimension 1 with no subgroups of dimension strictly between 0 and 1?
I will use $dimA$ to denote the Hausdorff dimension of a set $A \subseteq \mathbb{R}$. Being a null set means having Lebesgue measure zero.
In the 1966 paper "Additive gruppen mit vorgegebener ...
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The symmetric group and the field with one element
I've heard a few times that the symmetric group is an algebraic group over a field with one element, and that the alternating group is quite specifically $SO_n(\mathbb{F}_1)$. This does make a lot of ...
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Who conjectured that a transitive projective plane is Desarguesian?
The only known finite projective plane with a transitive automorphism group is the Desarguesian plane $PG(2,q)$ and it seems likely that there are no others, although this is not (quite) proved.
...
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Some questions on unitarisability of discrete groups
In this post I would like to ask several of questions related to Dixmier problem. I will try to make the post as self-contained as possible.
A discrete group $G$ is unitarisable if for every Hilbert ...
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Number of finite index subgroups in a free abelian group
Let $n,m\in\mathbb{N}$. Is there a formula for the number of subgroups of index $n$ in $\mathbb{Z}^m$? Perhaps in terms of the divisors of $n$?
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Why is $\operatorname{SO}(4)$ not a simple Lie group?
$\DeclareMathOperator\SO{SO}$I'm not asking for a proof but for some hint that might be helpful to understand this "anomaly" in 4 dimensions. I'm aware of the parallelism with the $A_4$ ...
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What is a "block" in an abelian category?
In the literature and in some posts here, there has been variation in the undefined use of the term "block" for a category of modules over a ring, or more abstractly an abelian category (all of which ...
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Prove these are not surface groups
For $g,n \geq 1$, let $\Gamma_{g,n}$ be the group with the following presentation:
$$\langle \text{$a_1,b_1,\ldots,a_g,b_g$ $|$ $[a_1,b_1]^n [a_2,b_2] \cdots [a_g,b_g]=1$} \rangle.$$
For $n = 1$, ...
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Torsion-freeness of two groups with 2 generators and 3 relators and Kaplansky Zero Divisor Conjecture
Let $G_1$ and $G_2$ be the groups with the following presentations:
$$G_1=\langle a,b \;|\; (ab)^2=a^{-1}ba^{-1}, (a^{-1}ba^{-1})^2=b^{-2}a, (ba^{-1})^2=a^{-2}b^2 \rangle,$$
$$G_2=\langle a,b \;|\; ...
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Number of commuting pairs (triples, n-tuples) in GL_n(F_q) (and other groups)?
Question 1 What is the number of pairs of commuting elements in GL_n(F_q) ?
I am aware of many results concerning commuting elements in Mat_n(F_q), but I am interested in GL i.e. non-degenerate ...
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Is $ord(xy)$ independent of $ord(x)$ and $ord(y)$ in a finite group?
Let $r,s,t>1$ be positive integers. Must there exist a finite group $G$ with elements $x$ and $y$ such that $ord(x)=r$, $ord(y)=s$, and $ord(xy)=t$?
The answer is probably "yes." Is there a nice ...
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What is the universal property of the Weyl group?
If $G$ is a group and $H\le G$ a subgroup, let $NH$ denote the normalizer of $H$ in $G$, and let $WH = NH/H$; following May's Concise Course, §3.4 this I call the Weyl group. I have also seen the ...
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How should I formalize that there’s no canonical isomorphism between a finite Abelian group and its Pontryagin dual?
The title says it all, but let me repeat.
We all learn that the Pontryagin dual of a finite Abelian group is abstractly isomorphic as groups, but there’s no canonical isomorphism.
I think I ...
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"Big" groups $G$ with trivial $Out(G)$
We are looking for examples of groups $G$ such that $G$ is "big", but $Out(G)$ is trivial. By "big" we mean things like virtually free, or large, or Golod-Shafarevich. However, we would like our ...
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Determining if a matrix is orthogonal
Let g be an element of $GL_n(\mathbb C)$. We know that there are orthogonal groups $O(\beta)=\{X\in GL_n(\mathbb C) \mid X^t\beta X=\beta\}$ for any $\beta$, invertible symmetric matrix. Though these ...
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subsets of groups which have to be closed no matter what
One example of a subset of a group $G$ which has to be closed in any topology on $G$ compatible with the group operations is a centraliser. Are there any other interesting examples?
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The set of orders of elements in a group
Let $A$ be a subset of natural numbers. Consider the following problem:
Is there a group $G$ such that $\lbrace O(x) \; | \; x \in G \rbrace = A\cup\lbrace 1\rbrace$ ? (where $O(x)$ is the order of $...
user30230
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Can we bound degrees of complex irreps in terms of the average conjugacy class size?
This question arises when looking at a certain constant associated to (a certain Banach algebra built out of) a given compact group, and specializing to the case of finite groups, in order to try and ...
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Has dynamics on $G/\Gamma$ ever been used to prove interesting things about $\Gamma$?
Fix a Lie group $G$ and a discrete subgroup $\Gamma \subset G$. Homogeneous dynamics is about studying the actions of subgroups $H \subset G$ on the quotient $G/\Gamma$.
Does anyone know of an ...
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Realizing symmetric groups by diffeomorphisms
Let $M$ be a (closed, smooth) manifold of dimension $d$. For $n$ a positive integer, fix $n$ points $x_1, \dots, x_n \in M$. The group of diffeomorphisms of $M$ that permutes the points $x_i$ surjects ...
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Sylow theorems for infinite groups
Are there classes of infinite groups that admit Sylow subgroups and where the Sylow theorems are valid?
More precisely, I'm looking for classes of groups $\mathcal{C}$ with the following properties:
...
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Abelianization of general linear group of a polynomial ring
For $K$ a field, is it known what the abelianization of $GL_2(K[X])$ is?
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If each strict subgroup of G is free, must G be free or cyclic of prime order ?
If each strict subgroup of a group G is free, must G be free or cyclic of prime order ?
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Probability of commutation in a compact group
It is well known that if $G$ is a finite group, then the probability that two elements commutte is either $1$ (if $G$ is abelian) or less than or equal to $\frac58$.
If instead $K$ is a compact group,...
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Zero divisor conjecture and idempotent conjecture
Let $G$ be a torsion-free group and $C$ the ring of complex numbers. The zero divisor (idempotent, resp.) conjecture is that there is no nontrivial zerodivisor (idempotent, resp.) in $CG$.
The wiki ...
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Cosets and conjugacy classes
I'm interested in the following situation:
$G$ is a finite group;
$C$ is a conjugacy class in $G$;
$H$ is the centralizer of an element $h$ of $C$.
I want information on $|C\cap Hg|$ as $g$ varies ...
- 11.2k
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counter example for semi direct product of groups
Hi,
I have got a very natural question in group theory.
Suppose you have two countable groups $G_1,G_2$, some action of $\mathbb Z$ on them such that
the semi direct products are isomorphic $\phi:...
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Is it possible for a direct product to be isomorphic to the Zappa–Szép product?
Let $A$ and $G$ be two groups. Let $\alpha:G\rightarrow\operatorname{Bij}(A)$ be a group homomorphism and $\beta: A\rightarrow\operatorname{Bij}(G)$ an anti-homomorphism satisfying some conditions ...
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Which groups are Galois over some p-adic field?
Suppose I have some finite $p$-group $G$, or a little extension of it.
How do I know if there exists a prime $l$ and a finite extension $K$ of $\mathbb{Q}_l$ such that $G$ is the Galois group of ...
- 11.3k
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Kernel of linear representation of Baumslag-Solitar group
Let $BS(m,n)$ be the Baumslag-Solitar group defined by $B(m,n) = < a,b ~|~ b a^m b^{-1} = a^n > $, $mn \neq 0$. There is a linear representation of $BS(m,n)$ by mapping $a$ to the matrix $\left(...
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Number of positions of Rubik's cube grows with multiplier 13 with the distance - what are explanations and groups with similar growth pattern?
Rubik's cube and its generalizations attracts certain attention of mathematical community. It is somehow "noteworthy" that it has been proved that diameter of the Rubik's cube group is 20, i.e. ...
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Normal subgroups of finite index in free groups
Hi all,
This is a question about the groups $H_{n,s}$ introduced by Völklein in his book "Groups as Galois groups", §7.1, and defined as follows: let $N$ be the intersection of all normal subgroups ...
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Topological semi-direct products of groups
In Kaniuth, Taylor, Induced representations of locally compact groups on pages 9-10 it's claimed that if $G$ is a locally compact group with closed subgroups $N,H$, with $N$ normal in $G$, with $N\cap ...
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Linear occurrences of finite simple groups
Let $S$ be a finite simple group. All representations below are over the complex numbers.
Let
$d_0(S)$ be the smallest dimension of a faithful representation of $S$,
$d_1(S)$ be the smallest ...
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Is group theory useful in any way to optimization?
For what I have seen, optimization uses a lot of linear algebra and convex analysis, but I have not seen any group theory being used, so I was curious about it.
Is group theory useful in any way to ...
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Orthogonal Groups over finite fields
Hello
Let $\mathbb{F}_p$ be the finite field with $p$ elements. One can show that over finite fields, there are just two non-degenerate quadratic forms.
So here I want to pick
any non-degenerate ...
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Action of SL(2,Z) on upper triangular primitive integer matrices of determinant N, from the right. Is it transitive?
I am porting this question across from StackExchange, since it has received no answers and perhaps is sufficiently deep to fit here.
I am considering the set of upper triangular matrices
$$D_N=\left\...
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Explicit construction of an element of ${\rm GL}(2, p)$ of order $p+1$
It is well-known that the order of $GL(2, p)$ is $(p^2-1)(p^2-p) = (p-1)^2(p+1)p.$
It is easy to construct matrices of orders $(p-1)$ and $p$ (diagonal and parabolic, respectively), but the only way ...
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Is this a well-known bound for the derived length of a finite group?
Let $cd(G)$ be the set of degrees of the irreducible complex characters of the finite group $G$.
It is conjectured that if $G$ is solvable then $dl(G)\leq |cd(G)|$ and it is a result by Gluck that $...
- 2,508
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Does a fibre product of a group $G$ with itself have a subgroup isomorphic to $G$?
Let $G$ be a group, and consider a fibre product of the form $H=G\times_{D,\phi,\psi}G$, i.e. the group of pairs $(g,g'),\phi(g)=\psi(g')$, for some surjective group morphisms $\phi:G\rightarrow D$ ...
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