Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
1,094 questions
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Conceptual reason why the sign of a permutation is well-defined?
Teaching group theory this semester, I found myself laboring through a proof that the sign of a permutation is a well-defined homomorphism $\operatorname{sgn} : \Sigma_n \to \Sigma_2$. An insightful ...
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How feasible is it to prove Kazhdan's property (T) by a computer?
Recently, I have proved that Kazhdan's property (T) is theoretically provable
by computers (arXiv:1312.5431,
explained below), but I'm quite lame with computers and have
no idea what they actually can ...
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Non-amenable groups with arbitrarily large Tarski number?
Just out of curiosity, I wonder whether there are non-amenable groups with arbitrarily large Tarski numbers. The Tarski number $\tau(G)$ of a discrete group $G$ is the smallest $n$ such that $G$ ...
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Is there a 0-1 law for the theory of groups?
Several months ago, Dominik asked the question Is there a 0-1 law for the theory of groups? on mathstackexchange, but although his question received attention there is still no answer. By asking the ...
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Heuristic argument that finite simple groups _ought_ to be "classifiable"?
Obviously there exists a list of the finite simple groups, but why should it be a nice list, one that you can write down?
Solomon's AMS article goes some way toward a historical / technical ...
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Is Thompson's Group F amenable?
Last year a paper on the arXiv (Akhmedov) claimed that Thompson's group $F$ is not amenable, while another paper, published in the journal "Infinite dimensional analysis, quantum probability, and ...
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Fantastic properties of Z/2Z
Recently I gave a lecture to master's students about some nice properties of the group with two elements $\mathbb{Z}/2\mathbb{Z}$. Typically, I wanted to present simple, natural situations where the ...
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Is there an odd-order group whose order is the sum of the orders of the proper normal subgroups?
For a finite group G, let |G| denote the order of G and write $D(G) = \sum_{N \triangleleft G} |N|$, the sum of the orders of the normal subgroups. I would like to call G "perfect" if D(G) = 2|G|, ...
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a categorical Nakayama lemma?
There are the following Nakayama style lemmata:
(the classical Nakayama lemma) Let $R$ be a commutative ring with $1$ and $M$ a finitely generated $R$-module. If $m_1, \ldots, m_n$ generate $M$ ...
user19475
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Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?
$\DeclareMathOperator\SO{SO}\newcommand{\R}{\mathbb{R}}\newcommand{\S}{\mathbb{S}}$The periodic table of elements has row lengths $2, 8, 8, 18, 18, 32, \ldots $, i.e., perfect squares doubled. The ...
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Which philosophy for reductive groups?
I am just beginning to look further into trace formulas and automorphic forms in a quite general setting. For long I have noticed that the natural assumption on the group $G$ we work on is to be ...
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Can the unsolvability of quintics be seen in the geometry of the icosahedron?
Q1. Is it possible to somehow "see" the unsolvability of quintic polynomials
in the $A_5$ symmetries of the icosahedron (or dodecahedron)?
Perhaps this is too vague a question.
Q2. Are there ...
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What are some interesting corollaries of the classification of finite simple groups?
The classification of finite simple groups, whether it be viewed as finished, or as a work in progress, is (or will be) without doubt an enormous achievement. It clearly sheds a great deal of light on ...
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Are semi-direct products categorical (co)limits?
Products, are very elementary forms of categorical limits. My question is whether in the category of groups, semi-direct products are categorical limits.
As was pointed in:
http://unapologetic....
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(co)homology of symmetric groups
Let $S_n=\{\text{bijections }[n]\to[n]\}$ be the n-th symmetric group. Its (co)homology will be understood with trivial action. What are the $\mathbb{Z}$-modules $H_k(S_n;\mathbb{Z})$? Using GAP, we ...
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Order of products of elements in symmetric groups
Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying
$1 < a, b, c \leq n-2$ the symmetric group ${\rm S}_n$ has elements of order $a$ and $b$
whose product has order $...
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answer
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Can one explain Tannaka-Krein duality for a finite-group to ... a computer ? (How to make input for reconstruction to be finite datum?)
Consider a finite group. Tannaka-Krein duality allows to reconstruct the group from the
category of its representations and additional structures on it (tensor structure + fiber functor). Somehow ...
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Is it decidable whether or not a collection of integer matrices generates a free group?
Suppose we have integer matrices $A_1,\ldots,A_n\in\operatorname{GL}(n,\mathbb Z)$. Define $\varphi:F_n\to\operatorname{GL}(n,\mathbb Z)$ by $x_i\mapsto A_i$. Is there an algorithm to decide whether ...
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Realizing groups as automorphism groups of graphs.
Frucht showed that every finite group is the automorphism group of a finite graph. The paper is here.
The argument basically is that a group is the automorphism group of its (colored) Cayley graph
...
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Does GL_n(Z) have a noetherian group ring?
Has the (left, right, 2-sided) noetherian property of the integral group ring of arithmetic groups like $GL_n(Z)$ been considered in the literature?
Motivation: a recent trend has been to study "...
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On the size of balls in Cayley graphs
Next semester I will be teaching an introductory course on geometric group theory and there is a basic question that I do not know the answer to. Let $G$ be a finitely generated group with finite ...
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What relationship, if any, is there between the diameter of the Cayley graph and the average distance between group elements?
It's known that every position of Rubik's cube can be solved in 20 moves or less. That page includes a nice table of the number of positions of Rubik's cube which can be solved in k moves, for $k = 0,...
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answers
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The number of polynomials on a finite group
A function $f:X\to X$ on a group $X$ is called a polynomial if there exist $n\in\mathbb N=\{1,2,3,\dots\}$ and elements $a_0,a_1,\dots,a_n\in X$ such that $f(x)=a_0xa_1x\cdots xa_n$ for all $x\in X$. ...
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Faithful representations and tensor powers
The following result was mentionned earlier in this thread, I searched a bit in the related threads and couldn't find a proof. I would really like to see a proof of it:
Let $G$ be a finite group and $...
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maximal order of elements in GL(n,p)
I am looking for a formula for the maximal order of an element in the group $\operatorname{GL}\left(n,p\right)$, where $ p$ is prime.
I recall seeing such a formula in a paper from the mid- or early ...
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What is the largest Laver table which has been computed?
Richard Laver proved that there is a unique binary operation $*$ on $\{1,\ldots,2^n\}$ which satisfies $$a*1 \equiv a+1 \mod 2^n$$
$$a* (b* c) = (a* b) * (a * c).$$
This is the $n$th Laver table $(A_n,...
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Cogroup objects
Pretty much anyone who does algebra is familiar with group objects in categories, but what about cogroup objects? Most of what I've been able to find about them is that they "arise naturally in ...
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Isomorphism of $\mathbb{Z}\ltimes_A \mathbb{Z}^m$ and $\mathbb{Z}\ltimes_B \mathbb{Z}^m$
here it's a question that I've posted in MSE but unfortunately got no answers:
Let $A$ and $B$ be matrices of finite order with integer coefficients.
Let $n\in\mathbb{N}$ and let $G_A=\mathbb{Z}\...
20
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7
answers
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Understanding groups that are not linear
I have a really hard time "feeling" what it means for a group to fail to be linear. Vaguely, I'd like to know how one should think about such groups. More precisely:
What are some interesting ...
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Find a "natural" group that contains the quotient of the infinite symmetric group by the alternating subgroup
Let $S_\infty$ the group of permutations of $\mathbb{N}$. It can be shown that there is no homomorphism $S_\infty \to \mathbf{Z}/2$ extending the sign on the finite symmetric groups. Is it possible to ...
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What is the outer automorphism group of SU(n)?
All the automorphisms of $SU(2)$ seem to be inner, which would mean that $\mathrm{Out}$ $SU(2)$ is trivial. Is that correct? Is this true in general $SU(n)$? I can't quite see -- any thoughts would be ...
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How to add two numbers from a group theoretic perspective?
It is known that adding two numbers and looking at the carrying operation has a link with cocycles in group theory. (https://www.jstor.org/stable/3072368?origin=crossref)
When we add two numbers by ...
user6671
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Number of isomorphism types of finite groups
Are there some good asymptotic estimations for the number $F(n)$ of non-isomorphic finite groups of size smaller than $n$?
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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
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God's number for the $n \times n \times n$-cube
This is a question about Rubik's Cube and generalizations of this puzzle, such as Rubik's Revenge, Professor's cube or in general the $n \times n \times n$ cube.
Let $g(n)$ be the smallest number $m$, ...
- 63.1k
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No injective groups with more than one element?
There are several claims in the literature that there are no injective groups (with more than one element), but I have not found a proof. For example, Mac Lane claims in his Duality from groups paper ...
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Number of conjugacy classes in GL(n,Z)
Like every other group also $\mbox{GL}(n,\mathbb{Z})$ acts on the set of all its subgroups, by conjugation: if $\phi \in \mbox{GL}(n,\mathbb{Z})$, then $\phi$ acts by $H \mapsto \phi H \phi^{-1}$, ...
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Folner sets and balls
Several related questions were asked before on MO, but it is not clear to me if the following was settled.
Given a finitely generated amenable group, is it always possible to find some finite ...
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Finite groups with elements of the same order
Given a finite group $G$, let $\{(1,1),(m_1,n_1),\ldots,(m_r,n_r)\}$ be the list of pairs $(m,n)$ in which $m$ is the order of some element, and $n$ is the number of elements with this order. The ...
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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&...
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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 ...
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For what sets $X$ do there exist a pair of functions from $X$ to $X$ with the identity being the only function that commutes with both?
It is not too difficult to show that if $X$ is an infinite set, then there exists a two-element subset of the group $\operatorname{Sym}(X)$ with trivial centralizer iff $\lvert X\rvert \leq \lvert\...
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Possible orders of products of 2 involutions which interchange disjoint residue classes of the integers
Definition / Question
Definition: 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
$...
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Is each finite group multifactorizable?
Definition. A finite group $G$ is called multifactorizable if for any positive integer numbers $a_1,\dots,a_n$ with $a_1\cdots a_n=|G|$ there are subsets $A_1,\dots,A_n\subset G$ such that $A_1\cdots ...
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A group theoretic interpretation of Lagarias inequality
Let $G$ be a finite group, $S \subset G$ a generating set. Set $\sigma(G):=\sum_{U \subset G} |U| $, where the sum runs over all subgroups $U$ of $G$. Set $H_G := \sum_{g \in G} \frac{1}{|g|+1}$, ...
user6671
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1
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Homomorphisms from powers of Z to Z
I believe it is known that if I is a set of non-measurable cardinality, then any homomorphism $Z^I\to Z$ factors through a finite power. Here $Z$ is the group of integers. Can anyone give a ...
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Automorphism groups of odd order
This is inspired by this question. Is there a description of finite groups without automorphisms of order $2$?
user6976
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1
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CAT(0) groups that does not act on CAT(0) cubical complex
CAT(0) groups are groups that act on a CAT(0) space properly and cocompactly. If a group acts on a CAT(0) cubical complex properly and cocompactly, then of course it is a CAT(0) Group. I am wondering ...
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How are reflection groups related to general point groups?
I always tried to understand how the finite reflection groups of $\Bbb R^d$ (of some fixed dimension $d$) relate to the point groups of the same space $\smash{\Bbb R^d}$ (finite subgroup of the ...
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Chevalley Groups over an arbitrary ring.
My question is simply about the Chevalley groups over rings. In many books, including Carter's book on "Simple groups of Lie types", the groups are considered over fields. I have checked the ...
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