All Questions
1,132 questions
87
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 ...
- 63.1k
13
votes
2
answers
2k
views
Generalization of a theorem of Øystein Ore in group theory
Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive.
Proof: see below.
Let $(H \subset G)$ be an inclusion of finite groups and $\...
50
votes
6
answers
11k
views
Generating finite simple groups with $2$ elements
Here is a very natural question:
Q: Is it always possible to generate a finite simple group with only $2$ elements?
In all the examples that I can think of the answer is yes.
If the answer is ...
- 7,609
174
votes
7
answers
17k
views
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))....
- 13k
113
votes
2
answers
16k
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 ...
- 3,511
42
votes
8
answers
11k
views
The finite subgroups of SU(n)
This question is inspired by the recent question "The finite subgroups of SL(2,C)". While reading the answers there I remembered reading once that identifying the finite subgroups of SU(3) is still an ...
- 2,123
42
votes
7
answers
10k
views
Bijection between irreducible representations and conjugacy classes of finite groups
Is there some natural bijection between irreducible representations and conjugacy classes of finite groups (as in case of $S_n$)?
- 1,318
94
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}$?
- 63.1k
13
votes
2
answers
1k
views
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 ...
- 24.9k
79
votes
12
answers
13k
views
Is there a high-concept explanation for why characteristic 2 is special?
The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, ...
- 118k
66
votes
2
answers
9k
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 ...
- 35.5k
43
votes
7
answers
8k
views
Why are free groups residually finite?
Why is it that every nontrivial word in a free group (it's easy to reduce to the case of, say, two generators) has a nontrivial image in some finite group? Equivalently, why is the natural map from a ...
- 2,356
7
votes
3
answers
698
views
Jordan-Hölder theorem for subfactors?
All the subfactors $(N\subset M)$ are irreducible and finite index inclusions of II$_1$ factors.
First recall that in this paper, D. Bisch characterizes the Jones projections $e_K$ of the ...
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 ...
- 44.8k
35
votes
2
answers
3k
views
Examples of finite groups with "good" bijection(s) between conjugacy classes and irreducible representations?
For symmetric group conjugacy classes and irreducible representation both are parametrized by Young diagramms, so there is a kind of "good" bijection between the two sets. For general finite groups ...
- 24.9k
20
votes
1
answer
587
views
$q$-(and other)-analogs for counting index-$n$ subgroups in terms of Homs to $S_n$?
The following formula of astonishing beauty and power (imho):
$$ \sum_{n \ge 0} \frac{| \mathrm{Hom}(G,S_n) | }{n! } z^n = \exp\left( \sum_{n \ge 1} \frac{|\text{Index}~n~\text{subgroups of}~ G|}nz^...
- 24.9k
17
votes
5
answers
7k
views
General bound for the number of subgroups of a finite group
I am interested in the following:
Let $G$ be a finite group of order $n$. Is there an explicit function $f$ such that
$|s(G)| \leq f(n)$ for all $G$ and for all natural numbers $n$, where $s(G)$ ...
user13040
12
votes
3
answers
1k
views
Is there a purely group-theoretic reformulation of an equivalence of subgroups?
There is an equivalence relation between inclusion of finite groups coming from the world of subfactors:
Definition: $(H_{1} \subset G_{1}) \sim(H_{2} \subset G_{2})$ if $(R^{G_{1}} \subset R^{H_{1}}...
64
votes
4
answers
8k
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 ...
- 1,065
36
votes
1
answer
3k
views
Whence “homomorphism” and “homomorphic”?
Today homomorphism (resp. isomorphism) means what Jordan (1870) had called isomorphism (resp. holoedric isomorphism). How did the switch happen?
“Homomorphic” (and “homomorphism” as “property of being ...
- 31.5k
28
votes
1
answer
2k
views
Have finite doubly transitive groups been classified?
I am trying to determine whether the literature contains a complete proof of the classification of finite 2-transitive groups. This is a fundamental result with important applications in many areas ...
- 6,397
23
votes
4
answers
3k
views
Non-vanishing of group cohomology in sufficiently high degree
Atiyah in his famous paper , Characters and cohomology of finite groups, after proving completion of representation ring in augmentation ideal is the same as $ K(BG)$, gives bunch of corollaries of ...
- 1,003
23
votes
2
answers
2k
views
Orbit structures of conjugacy class set and irreducible representation set under automorphism group
let G be a finite group. Suppose C is the set of conjugacy classes of G and R is the set of (equivalence classes of) irreducible representations of G over the complex numbers.
The automorphism group ...
- 7,320
17
votes
0
answers
969
views
Groups generated by 3 involutions
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
$\tau_{r_1(m_1),r_2(m_2)}$ be the ...
- 19.6k
13
votes
1
answer
2k
views
A dual version of a theorem of Øystein Ore in group theory
This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved:
Theorem: Let $[H, G]$ be a distributive interval of finite groups. Then $\exists g \in G$ such ...
11
votes
1
answer
717
views
Factorization of a finite group by two subsets
I want to write a GAP program for checking the following question.
Let $G$ be a given finite group with order $n$. Is it true that for every factorization $n=ab$ there exist subsets $A$ and $B$ ...
- 995
9
votes
2
answers
676
views
Powers of finite simple groups
I have heard about the following result: for each finite simple non-abelian group $S$ and each natural number $r\ge 2$ there exists a number $n=n(r,S)$ such that the power $S^n$ is $r$-generator but $...
- 331
3
votes
0
answers
303
views
Growth functions of finite group - computation, typical behaviour, surveys?
Looking on the growth function for Rubik's group and symmetric group, one sees rather different behaviour:
Rubik's growth in LOG scale (see MO322877):
S_n n=9 growth and nice fit by normal ...
- 24.9k
0
votes
2
answers
202
views
Products of maximal inclusions of finite groups with a non-obvious intermediate
Let $(H_1 \subset G_1)$ and $(H_2 \subset G_2)$ be core-free maximal inclusions of finite groups.
Their product, the inclusion $(H_1 \times H_2 \subset G_1 \times G_2)$, admits four obvious ...
78
votes
3
answers
10k
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 ...
- 22.8k
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\...
- 17.5k
47
votes
2
answers
4k
views
Collapsible group words
What is the length $f(n)$ of the shortest nontrivial group word $w_n$ in $x_1,\ldots,x_n$ that collapses to $1$ when we substitute $x_i=1$ for any $i$?
For example, $f(2)=4$, with the commutator $[...
- 23.8k
46
votes
2
answers
8k
views
Definition of "finite group of Lie type"?
The list of finite simple groups of Lie type has been understood for half a century, modulo some differences in notation (and identifications between some of the very small groups coming from ...
- 52.9k
43
votes
0
answers
2k
views
Why are there so few quaternionic representations of simple groups?
Having spent many hours looking through the Atlas of Finite Simple Groups while in Grad school, I recall being rather intrigued by the fact that among the sporadic groups, only one (McLaughlin as I ...
- 5,191
42
votes
6
answers
4k
views
Measures of non-abelian-ness
Let $G$ be a finite non-abelian group of $n$ elements.
I would like a measure that intuitively captures the
extent to which $G$ is non-commutative.
One easy measure is a count of the non-commutative ...
- 151k
36
votes
2
answers
5k
views
Smallest permutation representation of a finite group?
Given a finite group G, I'm interested to know the smallest size of a set X such that G acts faithfully on X. It's easy for abelian groups - decompose into cyclic groups of prime power order and add ...
- 3,612
35
votes
3
answers
4k
views
Is every abelian group a colimit of copies of Z?
More precisely, is every abelian group a colimit $\text{colim}_{j \in J} F(j)$ over a diagram $F : J \to \text{Ab}$ where each $F(j)$ is isomorphic to $\mathbb{Z}$?
Note that this does not follow ...
- 118k
33
votes
2
answers
1k
views
Analogies supporting heuristic: Weyl groups = algebraic groups over field with one element?
There is well-known heuristic that Weyl groups are reductive algebraic groups over "field with one element".
Probably the best known analogy supporting that heuristic is the limit $q\to1$ ...
- 24.9k
31
votes
1
answer
3k
views
Diameter of symmetric group
Let $\Sigma_n\subset G$ be a set of generators of the symmetric group $S_n$. It is a well-known conjecture that the diameter of the Cayley graph $\Gamma(S_n,\Sigma_n)$ is at most $n^C$ for some ...
- 20.2k
30
votes
2
answers
3k
views
In any Lie group with finitely many connected components, does there exist a finite subgroup which meets every component?
This question concerns a statement in a short paper by S. P. Wang titled “A note on free subgroups in linear groups" from 1981. The main result of this paper is the following theorem.
Theorem (Wang, ...
- 2,374
29
votes
4
answers
3k
views
Geometric interpretation of the lower central series for the fundamental group?
For any group $G$ we can form the lower central series of normal subgroups by taking $G_0 = G$, $G_1 = [G,G]$, $G_{i+1} = [G,G_i]$. We can check this gives a normal chain
$$G_0 \ge G_1 \ge ... \ge G_i ...
- 573
27
votes
4
answers
2k
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Units in the group ring over fours group after Gardam
Giles Gardam recently found (arXiv link) that Kaplansky's unit conjecture fails on a virtually abelian torsion-free group, over the field $\mathbb{F}_2$.
This conjecture asserted that if $\Gamma$ is a ...
- 6,652
19
votes
7
answers
3k
views
Universal cover of SL2(R) admits no central extensions?
Is it true that the universal cover of $\mathrm{SL}_2(\mathbb{R})$ has no non-trivial central extensions... as an abstract group?
(that's certainly true as a Lie group)
Motivation:
I have a projective ...
- 43.2k
14
votes
2
answers
3k
views
Example of Noetherian group (every subgroup is finitely generated) that is not finitely presented
A Noetherian group (also sometimes called slender groups) is a group for which every subgroup is finitely generated. (Equivalently, it satisfies the ascending chain condition on subgroups).
A ...
- 7,320
14
votes
2
answers
749
views
Solving the Bring quintic using the Monster?
I. Method
Hermite's method to solve the Bring quintic by functions that obey $x^8+y^8=1$ implicitly uses octahedral symmetry, while Emil Jann Fiedler's solution by the Rogers-Ramanujan continued ...
- 12.6k
12
votes
2
answers
2k
views
Do all exact sequences $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split for finitely generated abelian groups?
Suppose $A$ and $B$ are finitely generated Abelian groups. Are all exact sequences of the form $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split?
If not, is there an example?
- 307
8
votes
2
answers
2k
views
Maximal order of finite subgroups of $GL(n,Z)$
I am interested in the finite subgroups of $GL(n,Z)$ of maximal order.
Except for the dimensions $n = 2,4,6,7,8,9,10$ they are -- up to conjugacy in $GL(n,Q)$ -- in each dimension the group of signed ...
- 1,312
7
votes
1
answer
506
views
$G$ cocycle split to a coboundary in $J$, via a group extension
Consider a generic nontrivial $d$-cocycle $\omega_d^G \in H^d(G,U(1))$ in the cohomology group of a group $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. In otherwords, here the $d$-cocycle $\...
- 10.5k
5
votes
2
answers
505
views
A finiteness property for semi-simple algebraic groups
Let $G$ be a semi-simple algebraic group over a field $K$, I am considering a question about whether there exists a finite set of semi-simple $K$-subgroups, say $H_1,...,H_r$, such that for any semi-...
- 199
4
votes
0
answers
266
views
Metrics on finite groups and generalizations of central limit theorems for balls volumes (à la Diaconis-Graham)
In wonderful lectures by P. Diaconis "Group representations in probability and statistics, Chapter 6. Metrics on Groups, and Their Statistical Use" metrics on permutation groups are considered and ...
- 24.9k