Questions tagged [gr.group-theory]

Questions about the branch of algebra that deals with groups.

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171 votes
7 answers
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Does $\DeclareMathOperator\Aut{Aut}\Aut(\Aut(\dots\Aut(G)\dots))$ stabilize?

Purely for fun, I was playing around with iteratively applying $\DeclareMathOperator{\Aut}{Aut}\Aut$ to a group $G$; that is, studying groups of the form $$ {\Aut}^n(G):= \Aut(\Aut(\dots\Aut(G)\dots))....
Greg Muller's user avatar
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163 votes
28 answers
54k views

Cool problems to impress students with group theory [closed]

Since this forum is densely populated with algebraists, I think I'll ask it here. I'm teaching intermediate level algebra this semester and I'd like to entertain my students with some clever ...
160 votes
37 answers
15k views

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 ...
Tim Campion's user avatar
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114 votes
3 answers
5k views

The number $\pi$ and summation by $SL(2,\mathbb Z)$

Let $f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}$. (it is the defect in the triangle inequality) Then, we discovered by heuristic arguments and then verified by computer that $$\...
Nikita Kalinin's user avatar
111 votes
2 answers
15k views

Does every non-empty set admit a group structure (in ZF)?

It is easy to see that in ZFC, any non-empty set $S$ admits a group structure: for finite $S$ identify $S$ with a cyclic group, and for infinite $S$, the set of finite subsets of $S$ with the binary ...
Konrad Swanepoel's user avatar
108 votes
11 answers
12k views

Why do Groups and Abelian Groups feel so different?

Groups are naturally "the symmetries of an object". To me, the group axioms are just a way of codifying what the symmetries of an object can be so we can study it abstractly. However, this heuristic ...
Greg Muller's user avatar
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101 votes
4 answers
5k views

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 ...
Narutaka OZAWA's user avatar
91 votes
2 answers
7k views

$A$ is isomorphic to $A \oplus \mathbb{Z}^2$, but not to $A \oplus \mathbb{Z}$

Are there abelian groups $A$ with $A \cong A \oplus \mathbb{Z}^2$ and $A \not\cong A \oplus \mathbb{Z}$?
Martin Brandenburg's user avatar
88 votes
5 answers
10k views

When is $A$ isomorphic to $A^3$?

This is totally elementary, but I have no idea how to solve it: let $A$ be an abelian group such that $A$ is isomorphic to $A^3$. is then $A$ isomorphic to $A^2$? probably no, but how construct a ...
Martin Brandenburg's user avatar
87 votes
1 answer
9k views

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 $G$ ...
Narutaka OZAWA's user avatar
84 votes
23 answers
10k views

Solving algebraic problems with topology

Often, topologists reduce a problem which is - in some sense - of geometric nature, into an algebraic question that is then (partiallly) solved to give back some understanding of the original problem. ...
81 votes
4 answers
12k views

Can a group be a universal Turing machine?

This question was inspired by this blog post of Jordan Ellenberg. Define a "computable group" to be an at most countable group $G$ whose elements can be represented by finite binary strings, with the ...
Terry Tao's user avatar
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81 votes
3 answers
5k views

How do I verify the Coq proof of Feit-Thompson?

I probably don't have the appropriate background to even ask this question. I know next to nothing about formal or computer-aided proof, and very little even about group theory. And this question is ...
Nate Eldredge's user avatar
77 votes
3 answers
9k views

5/8 bound in group theory

The odds of two random elements of a group commuting is the number of conjugacy classes of the group $$ \frac{ \{ (g,h): ghg^{-1}h^{-1} = 1 \} }{ |G|^2} = \frac{c(G)}{|G|}$$ If this number exceeds ...
john mangual's user avatar
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76 votes
6 answers
9k views

Which graphs are Cayley graphs?

Every group presentation determines the corresponding Cayley graph, which has a node for each group element, and arrows labeled with the generators to get from one group element to another. My main ...
Joel David Hamkins's user avatar
75 votes
5 answers
3k views

When the automorphism group of an object determines the object

Let me start with three examples to illustrate my question (probably vague; I apologize in advance). $\mathbf{Man}$, the category of closed (compact without boundary) topological manifold. For any $M,...
74 votes
4 answers
5k views

Groups that do not exist

In the long process that resulted in the classification of finite simple groups, some of the exceptional groups were only shown to exist after people had computed (most of) their character tables and ...
Mariano Suárez-Álvarez's user avatar
73 votes
2 answers
7k views

The inverse Galois problem and the Monster

I have a slight interest in both the inverse Galois problem and in the Monster group. I learned some time ago that all of the sporadic simple groups, with the exception of the Mathieu group $M_{23}$, ...
aorq's user avatar
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72 votes
4 answers
4k views

Is ${\rm S}_6$ the automorphism group of a group?

The automorphism group of the symmetric group $S_n$ is $S_n$ when $n$ is not $2$ or $6$, in which cases it is respectively $1$ and the semidirect product of $S_6$ with the (cyclic) group of order $2$. ...
Benoit Jubin's user avatar
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70 votes
9 answers
16k views

Is there a slick proof of the classification of finitely generated abelian groups?

One the proofs that I've never felt very happy with is the classification of finitely generated abelian groups (which says an abelian group is basically uniquely the sum of cyclic groups of orders $...
Ben Webster's user avatar
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70 votes
9 answers
8k views

What are "classical groups"?

Unlike many other terms in mathematics which have a universally understood meaning (for instance, "group"), the term classical group seems to have a fuzzier definition. Apparently it originates with ...
Jim Humphreys's user avatar
70 votes
1 answer
5k views

Nontrivial finite group with trivial group homologies?

I stumbled across this question in a seminar-paper a long time ago: Does there exist a positive integer $N$ such that if $G$ is a finite group with $\bigoplus_{i=1}^NH_i(G)=0$ then $G=\lbrace 1\...
Chris Gerig's user avatar
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70 votes
2 answers
6k views

Group cohomology and condensed matter

I am mystified by formulas that I find in the condensed matter literature (see Symmetry protected topological orders and the group cohomology of their symmetry group arXiv:1106.4772v6 (pdf) by Chen, ...
Edward Witten's user avatar
69 votes
28 answers
7k views

Results from abstract algebra which look wrong (but are true)

There are many statements in abstract algebra, often asked by beginners, which are just too good to be true. For example, if $N$ is a normal subgroup of a group $G$, is $G/N$ isomorphic to a subgroup ...
67 votes
3 answers
4k views

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 ...
Seirios's user avatar
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66 votes
5 answers
6k views

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 ...
Tim Campion's user avatar
  • 60.6k
66 votes
1 answer
6k views

Why can't a nonabelian group be 75% abelian?

This question asks for intuition, not a proof. An earlier question, Measures of non-abelian-ness was thoroughly answered by Arturo Magidin. A paper by Gustafson1 proves that, for a nonabelian group, ...
Joseph O'Rourke's user avatar
64 votes
7 answers
14k views

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 ...
ADL's user avatar
  • 2,752
64 votes
2 answers
8k views

Where are the second- (and third-)generation proofs of the classification of finite simple groups up to?

According the the Wikipedia page, the second generation proof is up to at least nine volumes: six by Gorenstein, Lyons and Solomon dated 1994-2005, two covering the quasithin business by Aschbacher ...
David Roberts's user avatar
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63 votes
1 answer
4k views

Feit-Thompson conjecture

The Feit-Thompson conjecture states: If $p<q$ are primes, then $\frac{q^p-1}{q-1}$ does not divide $\frac{p^q-1}{p-1}$. On page xiii of these proceedings of a conference at the University of ...
Mare's user avatar
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62 votes
9 answers
9k views

Fundamental groups of noncompact surfaces

I got fantastic answers to my previous question (about modern references for the fact that surfaces can be triangulated), so I thought I'd ask a related question. A basic fact about surface topology ...
Andy Putman's user avatar
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62 votes
1 answer
4k views

Normalizers in symmetric groups

Question: Let $G$ be a finite group. Is it true that there is a subgroup $U$ inside some symmetric group $S_n$, such that $N(U)/U$ is isomorphic to $G$? Here $N(U)$ is the normalizer of $U$ in $S_n$. ...
Peter Mueller's user avatar
61 votes
3 answers
7k views

What is the current status of the Kaplansky zero-divisor conjecture for group rings?

Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group. A couple of good ...
Johan Öinert's user avatar
61 votes
5 answers
7k views

Why are abelian groups amenable?

A (discrete) group is amenable if it admits a finitely additive probability measure (on all its subsets), invariant under left translation. It is a basic fact that every abelian group is amenable. ...
Tom Leinster's user avatar
  • 27.2k
61 votes
1 answer
2k views

Are there $n$ groups of order $n$ for some $n>1$?

Given a positive integer $n$, let $N(n)$ denote the number of groups of order $n$, up to isomorphism. Question: Does $N(n)=n$ hold for some $n>1$? I checked the OEIS-sequence https://oeis.org/...
Peter's user avatar
  • 1,203
59 votes
4 answers
7k views

Is orientability a miracle?

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$This question is prompted by a recent highly-upvoted question, Conceptual reason why the sign of a permutation is well-defined? The responses made ...
Timothy Chow's user avatar
  • 78.1k
56 votes
6 answers
6k views

Can the symmetric groups on sets of different cardinalities be isomorphic?

For any set X, let SX be the symmetric group on X, the group of permutations of X. My question is: Can there be two nonempty sets X and Y with different cardinalities, but for which SX is isomorphic ...
Joel David Hamkins's user avatar
56 votes
14 answers
20k views

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 ...
55 votes
2 answers
6k views

How do you *state* the Classification of finite simple groups?

From the point of view of formal math, what would constitute an appropriate statement of the classification of finite simple groups? As I understand it, the classification enumerates 18 infinite ...
Mario Carneiro's user avatar
55 votes
3 answers
3k views

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|, ...
Tom Leinster's user avatar
  • 27.2k
55 votes
5 answers
2k views

Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?

Choose unit quaternions $q_0, q_1, q_2, q_3, q_4$ that form the vertices of a regular 4-simplex in the quaternions. Assume $q_0 = 1$. Let the other four generate a group via quaternion ...
John Baez's user avatar
  • 21.3k
54 votes
5 answers
10k views

Why are the sporadic simple groups HUGE?

I'm merely a grad student right now, but I don't think an exploration of the sporadic groups is standard fare for graduate algebra, so I'd like to ask the experts on MO. I did a little reading on them ...
REDace0's user avatar
  • 677
54 votes
4 answers
5k views

How many square roots can a non-identity element in a group have?

Let $G$ be a finite group. Let $r_2\colon G \to \mathbb{N}$ be the square-root counting function, assigning to each $g\in G$ the number of $x\in G$ with $x^2=g$. Perhaps surprisingly, $r_2$ does not ...
alpmu's user avatar
  • 785
54 votes
2 answers
2k views

How many relations of length $n$ can exists in a group without enforcing shorter relations?

Let $G$ be a group with two generators. Suppose that all non-trivial words of length less or equal $n$ in the generators and their inverses define non-trivial elements in $G$. Question: How many of ...
Andreas Thom's user avatar
  • 25.3k
53 votes
6 answers
5k views

Does homeomorphic and isomorphic always imply homeomorphically isomorphic?

Let $(G,\cdot,T)$ and $(H,\star,S)$ be topological groups such that $(G,T)$ is homeomorphic to $(H,S)$ and $(G,\cdot)$ is isomorphic to $(H,\star)$. Does it follow that $(G,\cdot,T)$ and $(H,\star,S)...
user avatar
53 votes
5 answers
5k views

How much of the ATLAS of finite groups is independently checked and/or computer verified?

In a recent talk Finite groups, yesterday and today Serre made some comments about proofs that rely on the classification of finite simple groups (CFSG) and on the ATLAS of Finite Groups. Namely, he ...
David Roberts's user avatar
  • 33.8k
53 votes
3 answers
7k views

Why are parabolic subgroups called "parabolic subgroups"?

Over the years, I have heard two different proposed answers to this question. It has something to do with parabolic elements of $SL(2,\mathbb{R})$. This sounds plausible, but I haven't heard a really ...
Timothy Chow's user avatar
  • 78.1k
52 votes
2 answers
3k views

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 $M$ ...
user avatar
52 votes
2 answers
5k views

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 ...
Eugene Stern's user avatar
52 votes
0 answers
988 views

Class function counting solutions of equation in finite group: when is it a virtual character?

Let $w=w(x_1,\dots,x_n)$ be a word in a free group of rank $n$. Let $G$ be a finite group. Then we may define a class function $f=f_w$ of $G$ by $$ f_w(g) = |\{ (x_1,\dots, x_n)\in G^n\mid w(x_1,\dots,...
Frieder Ladisch's user avatar

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