# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

304
questions

**12**

votes

**2**answers

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### 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 ...

**44**

votes

**6**answers

8k 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 ...

**33**

votes

**7**answers

7k 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 $S_n$)?

**41**

votes

**8**answers

9k 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 ...

**32**

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 ...

**38**

votes

**2**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, ...

**48**

votes

**2**answers

4k 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 ...

**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}}...

**23**

votes

**4**answers

5k 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 ...

**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 ...

**12**

votes

**1**answer

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### 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 ...

**20**

votes

**1**answer

465 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^...

**13**

votes

**2**answers

841 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 ...

**0**

votes

**2**answers

190 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

**1**answer

642 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 $B$ ...

**63**

votes

**9**answers

6k 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 ...

**36**

votes

**6**answers

3k 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 ...

**38**

votes

**2**answers

5k 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 ...

**14**

votes

**5**answers

5k 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 $s(G)$ ...

**9**

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 ...

**4**

votes

**0**answers

196 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 ...

**63**

votes

**10**answers

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### 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 ...

**53**

votes

**14**answers

14k 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 ...

**67**

votes

**3**answers

3k 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 ...

**30**

votes

**3**answers

4k 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 ...

**32**

votes

**9**answers

4k 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.
...

**22**

votes

**0**answers

957 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 ...

**15**

votes

**2**answers

1k 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 $n$?

**17**

votes

**0**answers

609 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 ...

**11**

votes

**1**answer

576 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 $H_G := \sum_{g \in G} \frac{1}{|g|+1}$, ...

**7**

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 ...

**12**

votes

**0**answers

573 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" ...

**2**

votes

**2**answers

581 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 versa?
...

**13**

votes

**3**answers

922 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 ...

**11**

votes

**1**answer

518 views

### Automorphism groups of odd order

This is inspired by this question. Is there a description of finite groups without automorphisms of order $2$?

**8**

votes

**2**answers

1k 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 ...

**5**

votes

**2**answers

562 views

### Factor subset of 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,...

**5**

votes

**0**answers

302 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 ...

**3**

votes

**1**answer

219 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) \...

**9**

votes

**2**answers

456 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 ...

**6**

votes

**1**answer

604 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 ...

**5**

votes

**1**answer

553 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 \...

**3**

votes

**1**answer

494 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 ...

**5**

votes

**1**answer

257 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) \...

**41**

votes

**10**answers

9k views

### The finite subgroups of SL(2,C)

Books can be written about the finite subgroups of $\mathrm{SL}(2,\mathbb C)$ (and their immediate family, like the polyhedral groups...) I am about to start writing notes for a short course about ...

**14**

votes

**2**answers

3k views

### learning Deligne-Lusztig theory

Can someone give me a roadmap for learning Deligne-Lusztig theory? (Except for the original article by Deligne and Lusztig)
Edit: You may assume knowledge of representation theory of finite groups (...

**64**

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,
...

**50**

votes

**3**answers

2k 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|, ...

**24**

votes

**3**answers

3k views

### Roots of permutations

Consider the equation $x^2=x_0$ in the symmetric group $S_n$, where $x_0\in S_n$ is fixed. Is it true that for each integer $n\geq 0$, the maximal number of solutions (the number of square roots of $...

**41**

votes

**3**answers

7k 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 ...