Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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93 votes
15 answers
32k views

Collecting proofs that finite multiplicative subgroups of fields are cyclic

I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic ...
79 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 ...
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 $2$. ...
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68 votes
10 answers
7k 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 ...
67 votes
20 answers
17k views

Fun applications of representations of finite groups

Are there some fun applications of the theory of representations of finite groups? I would like to have some examples that could be explained to a student who knows what is a finite group but does not ...
67 votes
10 answers
5k 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 ...
  • 1,988
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 ...
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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, ...
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 $S_n$. ...
60 votes
5 answers
5k 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 ...
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59 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/...
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53 votes
14 answers
17k 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 ...
53 votes
4 answers
4k 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 ...
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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 ...
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53 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|, ...
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53 votes
2 answers
6k 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 ...
  • 31.8k
49 votes
5 answers
9k 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 ...
  • 627
49 votes
1 answer
8k views

What is Atiyah's topological formulation of the odd order theorem?

Here is a quote from an article by Daniel Gorenstein on the history of the classification of finite simple groups (available here). During that year in Harvard, Thompson began his monumental ...
  • 2,576
49 votes
0 answers
923 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,...
48 votes
3 answers
2k views

Is each squared finite group trivial?

A semigroup $S$ is defined to be squared if there exists a subset $A\subseteq S$ such that the function $A\times A\to S$, $(x,y)\mapsto xy$, is bijective. Problem: Is each squared finite group ...
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48 votes
2 answers
5k 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 ...
48 votes
1 answer
8k views

Order of an automorphism of a finite group

Let G be a finite group of order n. Must every automorphism of G have order less than n? (David Speyer: I got this question from you long ago, but I don't know whether you knew the answer. I stil ...
  • 24.1k
46 votes
1 answer
3k views

Which small finite simple groups are not yet known to be Galois groups over Q?

The subject line pretty much says it all. To expand just a little bit: 1) What is the smallest simple group that is not yet known to occur as a Galois group over $\mathbb{Q}$? (Variants: not known ...
45 votes
6 answers
9k 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 ...
44 votes
1 answer
3k views

Do all exact $1 \to A \to A \times B \to B \to 1$ split for finite groups?

Let $A$, $B$ be finite groups. Is it true that all short exact sequences $1 \rightarrow A \rightarrow A \times B \rightarrow B \rightarrow 1$ split on the right? In other words, do there exist ...
44 votes
1 answer
4k views

Square roots of elements in a finite group and representation theory

Let $G$ be a finite group. In an an earlier question, Fedor asked whether the square root counting function $r_2:G\rightarrow \mathbb{N}$, which assigns to $g\in G$ the number of elements of $G$ that ...
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43 votes
3 answers
8k 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 ...
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 ...
  • 2,053
42 votes
1 answer
6k views

Has gnu(2048) been found?

The gnu (or Group NUmber) function describes how many groups there are of a given order. The number of groups of each order are known up to 2047, see https://www.math.auckland.ac.nz/~obrien/research/...
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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 ...
41 votes
2 answers
6k 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 ...
41 votes
3 answers
3k views

Names of finite groups

Question: If you have a finite group, how do you name it? If, for whatever reason, you have to list all subgroups of $GL_2({\mathbb F}_5)$ up to isomorphism in a paper, you are likely to write ...
40 votes
1 answer
1k views

Known and fixed gaps in the proof of the CFSG

As the "second-generation" proof of the Classification of Finite Simple Groups is being written up in the volumes by Gorenstein, Lyons, Aschbacher, Smith, Solomon, and others (see e.g. this ...
40 votes
1 answer
2k views

Orders of products of permutations

Let $p$ be a prime, $n\gg p$ not divisible by $p$ (say, $n>2^{2^p}$). Are there two permutations $a, b$ of the set $\{1,...,n\}$ which together act transitively on $\{1,2,...,n\}$ and such that all ...
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39 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 ...
39 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 ...
38 votes
3 answers
2k views

Do finite groups acting on a ball have a fixed point?

Suppose that $G$ is a finite group, acting via homeomorphisms on $B^n$, the closed $n$-dimensional ball. Does $G$ have a fixed point? A fixed point for $G$ is a point $p \in B^n$ where for all $g \...
  • 21.1k
38 votes
2 answers
1k views

Size of the smallest group not satisfying an identity.

Given $F = F(x_0,\ldots,x_n)$ the free group on $n+1$ generators. Define a function $M: F\rightarrow \mathbb{N}$ such that $F(w) = l$, if the smallest group in which $w$ is not an identity is of size ...
  • 481
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, ...
37 votes
2 answers
3k views

Why are Schur multipliers of finite simple groups so small?

Given a finite simple group $G$, we can consider the quasisimple extensions $\tilde G$ of $G$, that is to say central extensions which remain perfect. Some basic group cohomology (based on the ...
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37 votes
2 answers
2k views

A group-theoretic perspective on Frankl's union closed problem

Here is a group theoretic phrasing of a special case of the union closed conjecture: Question: Given a finite group $G$, is there an element of prime power order which is contained in at most half ...
36 votes
1 answer
2k views

Tell me an algebraic integer that isn't an eigenvalue of the sum of two permutations

Can you tell me an algebraic integer, with all archimedean absolute values less than 2, which is not an eigenvalue of $\pi_1 + \pi_2$ for any two permutation matrices $\pi_1,\pi_2$? Is it ...
  • 18.7k
36 votes
0 answers
849 views

Are there infinite versions of sporadic groups?

The classification of finite simple groups states roughly that every non-abelian finite simple group is either alternating, a group of Lie type, or a sporadic group. For each of the groups of Lie ...
  • 576
35 votes
4 answers
5k 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 even,...
  • 1,776
35 votes
7 answers
4k views

Applications of Frobenius theorem and conjecture

A theorem of Frobenius states that if $n$ divides the order of a finite group $G$, then the number of solutions to $x^n = 1$ in $G$ is a multiple of $n$. Frobenius conjectured that if the number of ...
35 votes
2 answers
1k views

Why do sporadic simple groups have so few conjugacy classes?

In finite group theory, there's a general intuition that the further away a group is from abelian, the fewer conjugacy classes it will have. So it is to be expected that non-abelian finite simple ...
  • 4,608
35 votes
1 answer
2k views

Order-increasing bijection from arbitrary groups to cyclic groups

In his answer to this previous MO question, Gjergji Zaimi referred to the statement that for every finite group $G$ of order $n$, there is a bijection $\sigma \colon G \to \mathbb{Z}/n\mathbb{Z}$ ...
  • 5,514
34 votes
7 answers
8k 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 $S_n$)?
  • 1,218
34 votes
5 answers
4k views

Character free proof that Frobenius kernel is a normal subgroup?

The question is in the title, but here is some background/reminders: A subgroup $H\neq\{1\}$ of a finite group $G$ is called a Frobenius complement if $H\cap H^g = \{1\}$ for all $g\in G\backslash H$....
  • 12.6k
34 votes
3 answers
1k views

Word evaluating to a group element and its inverse with different frequency

I'm supervising an undergraduate research project. Among other things, I've got the student to look at this paper of Gene Kopp and John Wiltshire-Gordon. This question arose from a missing complex ...

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