Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

Filter by
Sorted by
Tagged with
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 $\...
Sebastien Palcoux's user avatar
49 votes
6 answers
10k 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 ...
Hugo Chapdelaine's user avatar
42 votes
8 answers
10k 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 ...
Q.Q.J.'s user avatar
  • 2,083
39 votes
7 answers
9k 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 $S_n$)?
Dan's user avatar
  • 1,288
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
  • 33.4k
13 votes
2 answers
968 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 ...
Alexander Chervov's user avatar
12 votes
7 answers
1k views

Connection between cyclic group and exponential function

I have been thinking about this for a while, but now got to the point where I got stuck. I don't know if it might be considered as a research level question, but I would be very happy if somebody knew ...
user avatar
42 votes
3 answers
2k views

Are there "real" vs. "quaternionic" conjugacy classes in finite groups?

The complex irreps of a finite group come in three types: self-dual by a symmetric form, self-dual by a symplectic form, and not self-dual at all. In the first two cases, the character is real-valued, ...
Allen Knutson's user avatar
34 votes
2 answers
2k 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 ...
Alexander Chervov's user avatar
32 votes
4 answers
6k views

Classification of finite groups of isometries

Consider the problem of classifying the finite groups of isometries of $\mathbb{R}^n$. For $n=2$ it is cyclic and dihedral groups. For $n=3$ they are well known, probably from Kepler and are related ...
Mathieu Dutour Sikiric's user avatar
20 votes
1 answer
547 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^...
Alexander Chervov's user avatar
15 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 $s(G)$ ...
user avatar
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}}...
Sebastien Palcoux's user avatar
23 votes
2 answers
1k 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 ...
Vipul Naik's user avatar
  • 7,230
12 votes
1 answer
1k 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 ...
Sebastien Palcoux's user avatar
11 votes
1 answer
706 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 $B$ ...
M.H.Hooshmand's user avatar
0 votes
2 answers
197 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 ...
Sebastien Palcoux's user avatar
70 votes
10 answers
8k views

Is every finite group a group of "symmetries"?

I was trying to explain finite groups to a non-mathematician, and was falling back on the "they're like symmetries of polyhedra" line. Which made me realize that I didn't know if this was actually ...
Andrew McIntyre's user avatar
45 votes
2 answers
7k 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 ...
Jim Humphreys's user avatar
41 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 ...
Joseph O'Rourke's user avatar
22 votes
0 answers
1k views

Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?

If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...
William DeMeo's user avatar
10 votes
1 answer
1k views

Are there workable algebraic geometry approaches for the pentagon equation?

A pentagon equation is a system of polynomial equations of degree $3$ with several variables and integer coefficients, given by a fusion ring. A fusion ring is given by a finite set of integer ...
Sebastien Palcoux's user avatar
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 ...
eins6180's user avatar
  • 1,302
4 votes
3 answers
2k views

Representation theory of p-groups in particular upper tringular matrices over F_p

Finite p-groups - have p^n elements by definition. According to WP there is rich structure theory. Question: How far is representation theory of p-groups is understood? In case this question is too ...
Alexander Chervov's user avatar
4 votes
0 answers
253 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 ...
Alexander Chervov's user avatar
79 votes
12 answers
6k views

Why are characters so well-behaved?

Last year I attended a first course in the representation theory of finite groups, where everything was over C. I was struck, and somewhat puzzled, by the inexplicable perfection of characters as a ...
Saul Glasman's user avatar
  • 2,138
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
  • 2,361
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
  • 60.5k
56 votes
14 answers
19k 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
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
40 votes
6 answers
5k views

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 ...
32 votes
9 answers
5k views

How many groups of size at most n are there? What is the asymptotic growth rate? And what of rings, fields, graphs, partial orders, etc.?

Question. How many (isomorphism types of) finite groups of size at most n are there? What is the asymptotic growth rate? And the same question for rings, fields, graphs, partial orders, etc. ...
Joel David Hamkins's user avatar
32 votes
3 answers
3k 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 $...
Stefan Kohl's user avatar
  • 19.5k
31 votes
3 answers
6k 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 ...
Leo's user avatar
  • 1,579
17 votes
0 answers
678 views

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 ...
Alexander Chervov's user avatar
17 votes
2 answers
2k 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 $n$?
Al Tal's user avatar
  • 1,271
13 votes
3 answers
1k views

Characterization of Frobenius complements

I have learned that Frobenius complements are characterized (among finite groups) by having a fixed point free complex representation. That is, a finite group $G$ is a Frobenius complement if and only ...
Joonas Ilmavirta's user avatar
13 votes
0 answers
717 views

Bijection between conjugacy classes and irreducible representation of Weyl group = Langlands correspondence over "field with one element"

Context: The number of conjugacy classes equals to the number of irreducuble representations (over C) for any finite group. Moreover for the symmetric group there is well-known "natural bijection" ...
Alexander Chervov's user avatar
11 votes
1 answer
616 views

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 $H_G := \sum_{g \in G} \frac{1}{|g|+1}$, ...
user avatar
11 votes
1 answer
762 views

Automorphism groups of odd order

This is inspired by this question. Is there a description of finite groups without automorphisms of order $2$?
user avatar
9 votes
2 answers
517 views

Strongly real elements of odd order in sporadic finite simple groups

Recall that an element of a finite group is said to be real if it is conjugate to its inverse, and strongly real if the conjugating element can be chosen to be an involution. Question: Is it true ...
John Murray's user avatar
  • 1,070
8 votes
4 answers
2k views

Product of conjugacy classes - is there an analog of Tanaka-Krein reconstruction ?

Consider a finite group G. The product of conjugacy classes can be defined in natural way just by multiplying the representatives and counting multiplicities (see e.g. MO 62088). So we get ring with ...
Alexander Chervov's user avatar
6 votes
1 answer
621 views

Positivity of the alternating sum of indices for boolean interval of finite groups

Let $G$ be a finite group and $H$ a subgroup such that the interval $[H,G]$ is a boolean lattice. Let $L_1, \dots , L_n$ be the maximal subgroups of $G$ containing $H$. Let the alternative sum ...
Sebastien Palcoux's user avatar
5 votes
1 answer
679 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 \subset ...
Sebastien Palcoux's user avatar
5 votes
2 answers
672 views

Factor subsets of a finite group

Let $G$ be a group of order $n$ and $d$ a positive divisor of $n$. Is it true that there exists a subset $A$ of $G$ with $d$ elements and a subset $B$ such that $G=AB$ and $|AB|=|A||B|$ (equivalently,...
M.H.Hooshmand's user avatar
5 votes
1 answer
290 views

Existence of a multiplication bifunctor for the category of groups

For $\mathsf{Grp}$ the category of groups, a bifunctor $M: \mathsf{Grp} \times \mathsf{Grp}\to \mathsf{Grp}$ is a multiplication bifunctor if: $M(C_n,C_m) \simeq C_{nm}$, $M(C_1,G) \simeq M(G,C_1) \...
Sebastien Palcoux's user avatar
5 votes
0 answers
305 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 ...
Sebastien Palcoux's user avatar
4 votes
1 answer
235 views

Existence of an addition bifunctor for the category of groups

Let $\mathsf{Grp}$ be the category of groups. A bifunctor $A: \mathsf{Grp} \times \mathsf{Grp}\to \mathsf{Grp}$ is an addition bifunctor if: $A(C_n,C_m) \simeq C_{n+m}$, $A(C_0,G) \simeq A(G,C_0) \...
Sebastien Palcoux's user avatar
4 votes
2 answers
704 views

If all real conjugacy classes are strongly real, then all real irreps are "strongly real"(symmetric), true?

Question Is true that if all real conjugacy classes of a finite group are strongly real, then all its real irreducible representations (irreps) are "strongly real" (symmetric)? And vice ...
Alexander Chervov's user avatar
3 votes
1 answer
587 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 ...
Sebastien Palcoux's user avatar

1
2 3 4 5
8