Questions on group theory which concern finite groups.

**53**

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**3**answers

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### 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 ...

**52**

votes

**4**answers

2k 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 ...

**50**

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**4**answers

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### Is 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 $2$. ...

**50**

votes

**1**answer

3k 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,
...

**43**

votes

**9**answers

3k 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 ...

**42**

votes

**1**answer

5k 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 ...

**42**

votes

**1**answer

2k 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 ...

**41**

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**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 ...

**37**

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**2**answers

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### 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|, ...

**37**

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**0**answers

2k 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 $S_n$.
...

**36**

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**20**answers

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### 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 ...

**35**

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**5**answers

4k 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 ...

**34**

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**5**answers

3k 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 ...

**34**

votes

**1**answer

541 views

### Are there $n$ groups of order $n$ for some $n>1$?

Denote $N(n)$ : number of groups of order $n$
Does $N(n)=n$ hold for some $n>1$ ?
I checked the OEIS-sequence https://oeis.org/A000001 as well as the squarefree numbers in the range ...

**33**

votes

**1**answer

1k 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 $\sigma \colon G \to \mathbb{Z}/n\mathbb{Z}$ ...

**32**

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**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 ...

**32**

votes

**3**answers

666 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 ...

**32**

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 ...

**31**

votes

**6**answers

3k 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 ...

**30**

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**14**answers

4k 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 ...

**30**

votes

**1**answer

2k 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 ...

**30**

votes

**0**answers

554 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 ...

**29**

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**4**answers

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### 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 ...

**29**

votes

**7**answers

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### 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 ...

**29**

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**2**answers

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### 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 ...

**29**

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**3**answers

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 ...

**28**

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**10**answers

6k 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 ...

**28**

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**2**answers

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### How much of the ATLAS of finite groups is independently checked and/or computer verified?

In a recent talk 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 said that a proof that relied on the ...

**28**

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**2**answers

692 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 ...

**28**

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**3**answers

2k 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 ...

**27**

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**9**answers

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### 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.
...

**27**

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**3**answers

1k views

### When is $S_n \times S_m$ a subgroup of $S_p$?

I asked the following question on math.stackexchange several months ago:
Let $n,m,p>1$ be such that $S_n \times S_m \hookrightarrow S_p$. Does it imply that $p \geq n+m$?
Derek Holt gave a ...

**27**

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**3**answers

1k views

### Being a subgroup: proof by character theory

Let me first cite a theorem due to Frobenius:
Let $G$ be a finite group, with $H$ a proper subgroup ($H\ne (1)$ and $G$). Suppose that for every $g\not\in H$, we have $H\cap gHg^{-1}=(1)$. Then
...

**26**

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**5**answers

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### Does $S_4$ inject into $SL(2,R)$ for some commutative ring $R$?

$\newcommand{\Z}{\mathbf{Z}}$
Given a nice infinite collection of groups, for example the symmetric groups, one can ask whether any finite group is a subgroup of one of them. Of course any finite ...

**26**

votes

**1**answer

830 views

### Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$

Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle.
Assume that the eigenvalues of $A$ are included in a circle arc of ...

**26**

votes

**2**answers

1k 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, ...

**26**

votes

**2**answers

2k 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 ...

**25**

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**7**answers

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 ...

**25**

votes

**1**answer

1k views

### Do all exact 1 -> A -> AxB -> B -> 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 ...

**25**

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**2**answers

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### 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 ...

**24**

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**11**answers

8k views

### Lecture notes on representations of finite groups

Next term I am supposed to teach a course on representation of finite groups. This is a third year course for undegrads. I was thinking to use the book of Grodon James and Martin Liebeck ...

**24**

votes

**3**answers

787 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 ...

**24**

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**4**answers

2k 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 ...

**24**

votes

**2**answers

1k views

### Smallest n for which G embeds in $S_n$?

Question: Given a finite group $G$, how do I find the smallest $n$ for which $G$ embeds in $S_n$?
Equivalently, what is the smallest set $X$ on which $G$ acts faithfully by permutations?
This ...

**24**

votes

**2**answers

900 views

### Is $\varphi(n)/n$ the maximal portion of $n$-cycles in a degree $n$ group?

Let $G$ be a degree $n$ group, i.e., a subgroup of the symmetric group $S_n$. Let $p(G)$ be the number of $n$-cycles in $G$ divided by the size of $G$.
Examples:
If $G$ is a cyclic transitive ...

**24**

votes

**1**answer

945 views

### How does one compute invariants of certain Grassmannians inside the regular representation?

Barry Mazur and I have come across the question below, motivated by (but independent
of) issues regarding the Leopoldt conjecture.
Suppose that $\mathbf{C}$ is the complex numbers.
Let $H$ be a ...

**23**

votes

**5**answers

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 ...

**23**

votes

**1**answer

967 views

### Enumeration of a finite group

Let $G=\{g_1,g_2,...,g_n\}$ be a group with $e=g_1$ and $n$ is odd,
Set $$a_1=g_1$$
$$a_2=g_1g_2$$
$$a_3=g_1g_2g_3$$
$$a_n=g_1g_2...g_n$$
I am looking for example that all $a_i$ are different from ...

**23**

votes

**2**answers

982 views

### Realizing groups as commutator subgroups

What are the groups $X$ for which there exists a group $G$ such that $G' \cong X$?
My considerations:
$\bullet$ If $X$ is perfect we are happy with $G=X$.
$\bullet$ If $X$ is abelian then $G := X ...

**23**

votes

**1**answer

1k 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 ...