Questions about the branch of abstract algebra that deals with groups.

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52
votes
4answers
4k 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 ...
11
votes
2answers
1k 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 ...
5
votes
3answers
479 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 ...
119
votes
7answers
8k views

Does $\mathrm{Aut}(\mathrm{Aut}(…\mathrm{Aut}(G)…))$ 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 $$ {\Aut}^n(G):= \Aut(\Aut(...\Aut(G)...)) $$ ...
0
votes
2answers
160 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 ...
11
votes
3answers
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 ...
82
votes
2answers
7k 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 ...
39
votes
1answer
3k 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 ...
22
votes
2answers
1k 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 ...
25
votes
6answers
3k 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 ...
15
votes
0answers
641 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 ...
10
votes
2answers
897 views

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 $n$?
3
votes
1answer
547 views

Abelian subfactors, a relevant concept?

Through the questions below, this post asks whether the concept of abelian subfactor is relevant. Remark : here abelian qualifies an inclusion of II$_1$ factors $(N \subset M)$, $N$ is not an abelian ...
8
votes
1answer
753 views

A dual version of a theorem of Øystein Ore in group theory

Let $(H \subset G)$ be an inclusion of finite groups. This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved: Theorem: $\mathcal{L}(H\subset G)$ ...
14
votes
2answers
899 views

Orbit structures of conjugacy class set and irreducible representation set under automorphism group

let G be a [EDIT: 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 ...
15
votes
1answer
790 views

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 $H \mapsto \phi H \phi^{-1}$, ...
6
votes
2answers
523 views

Groups whose centralisers are finite

Let $G$ be an infinite group such that the centraliser of any non central element is finite (and bounded). Is there any structure theorem known about $G$ ? Such a group seems to be at the other ...
4
votes
0answers
280 views

Are the homogeneous single chain subfactors, Dedekind?

Background: See here and there. Recall that a subfactor is Dedekind if all its intermediate subfactors are normal. A subfactor $(N \subset M)$ is Homogeneous Single Chain (HSC) if its lattice ...
12
votes
0answers
422 views

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 ...
10
votes
1answer
387 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 $B$ ...
4
votes
1answer
272 views

Normal intermediate subgroup and normal core

Let $G$ be a finite group and $H$ a subgroup. The normal core of $H$ in $G$ is $core_G(H) := \bigcap_{g \in G}g^{-1}Hg$ Definition: $K$ is a normal intermediate subgroup of the inclusion $(H ...
2
votes
0answers
108 views

Minimal number of defining relators of a finite $p$-group on a minimal generating set

What is the state-of-art of the following question? Let $p$ be any prime number. For any finite $p$-group $G$, let $r_G$ denote the minimum number of defining relators in all presentations of $G$ ...
4
votes
3answers
212 views

Relatively free algebras in a variety generated by a single algebra

Suppose $A$ is an algebra of signature $\mathcal{L}$ and $V=Var(A)$ is the variety generated by $A$. I want to know is it possible to classify relatively free elements of $V$? As a special case, for a ...
2
votes
1answer
390 views

A second isomorphism theorem for the inclusions of groups

The usual second isomorphism theorem for groups is: let $G$ be a group, $S$ and $N$ subgroups with $N$ normal, then $SN$ is a subgroup of $G$, $S\cap N$ is a normal subgroup of $S$ and $SN/N \simeq ...
59
votes
1answer
3k 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 ...
37
votes
14answers
5k 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 ...
26
votes
7answers
5k 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 ...
24
votes
2answers
1k views

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 ...
63
votes
2answers
2k 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 ...
29
votes
5answers
3k 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 ...
29
votes
4answers
2k views

For which $n$ is there only one group of order $n$?

Let $f(n)$ denote the number of (isomorphism classes of) groups of order $n$. A couple easy facts: If $n$ is not squarefree, then there are multiple abelian groups of order $n$. If $n \geq 4$ is ...
26
votes
6answers
2k 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 ...
23
votes
2answers
2k views

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 ...
14
votes
4answers
2k views

Relationship between the cohomology of a group and the cohomology of its associated Lie algebra.

Let $G$ be a group and let $k$ be a field (characteristic 0 if you want). Let $L$ be the graded Lie ring associated to the lower central series of $G$, that is, $L$, as a graded abelian group is ...
20
votes
2answers
2k views

Criteria for irreducibility of polynomial

If $f, g\in \mathbb C[a,b]$ are polynomials in two variables, are there easy criteria that allow to see if $f(x,y)-g(t,z)\in \mathbb C[x,y,t,z]$ is irreducible? Thank you very much, best
33
votes
2answers
1k 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 ...
22
votes
2answers
1k 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 ...
17
votes
2answers
1k views

(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 ...
29
votes
3answers
3k views

Feit-Thompson Theorem: The Odd Order Paper

For reference, the Feit-Thompson Theorem states that every finite group of odd order is necessarily solvable. Equivalently, the theorem states that there exist no non-abelian finite simple groups of ...
14
votes
5answers
1k 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 ...
10
votes
2answers
609 views

A group action of the Heisenberg group with special symmetries

Suppose we look at the Heisenberg group $H_{d}$ as a matrix group of upper triangular matrices over the ring $\mathbb{Z}/d\mathbb{Z}$. You can even choose $d$ to be prime if you want. A natural ...
28
votes
3answers
2k 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 ...
19
votes
4answers
2k 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 ...
22
votes
3answers
1k views

How can classifying irreducible representations be a “wild” problem?

Let $q$ be a prime power and $U_n(\mathbb{F}_q)$ be the group of unitriangular $n\times n$-matrices. I've read and heard in several places (see e.g. this mathoverflow question) that classifying ...
11
votes
4answers
1k views

Elements of infinite order in a profinite group

Say G is a profinite group with elements of arbitrarily large order. Do elements of infinite order exist (A) if we assume G is abelian? (B) in general? A start for (A): we can ask the same question ...
10
votes
5answers
3k views

infinite permutations

This question is related to this one: Continued fractions using all natural integers. Suppose we have the set of natural numbers $N$ with order and we perform permutation on it. So we obtain the same ...
8
votes
5answers
3k views

What are the normal subgroups of a direct product?

Let $N$ be a normal subgroup of $G \times H$, and let $\pi_1: G \times H \to G$ and $\pi_2: G \times H \to H$ be the canonical projections. Then $\pi_1(N)$ is normal in $G$ and $\pi_2(N)$ is normal in ...
6
votes
2answers
853 views

Hopfian property

Let $G$ be a group which is Hopfian and given a short exact sequence $1\to F \to H \to G \to 1$ with $F$ a finite normal subgroup of $H$. Is $H$ Hopfian?
28
votes
5answers
4k views

when is Aut(G) abelian

let $G$ be a group such that $Aut(G)$ is abelian. is then $G$ abelian? this is a sort of generalization of the well-known exercise, that $G$ is abelian when $Aut(G)$ is cyclic. but I have no idea. at ...
12
votes
3answers
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 ...