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97 votes
19 answers
38k 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 ...
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
  • 27.7k
47 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 ...
Pete L. Clark's user avatar
37 votes
1 answer
1k views

What is the smallest group not known to be a Galois group over $\mathbb{Q}$?

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$What is the smallest group not known to be a Galois group over $\mathbb{Q}$? Variants have been asked here before (e.g. Which small finite ...
Joachim König's user avatar
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 ...
JSE's user avatar
  • 19.2k
32 votes
3 answers
3k views

Is there a nice explanation for this curious fact about cyclic subgroups?

Here's something that I noticed that quite surprised me. Let $G$ be a finite abelian group. Consider the following expression. $$ \nu(G) = \sum_{\substack{H \leq G \\ H \text{ is cyclic}}} |H| $$ It ...
Simon Rose's user avatar
  • 6,290
31 votes
1 answer
2k views

Navigating $\mathbb{Z}/p\mathbb{Z}$

$\newcommand{\Z}{\mathbb{Z}}$Let's consider a silly-looking question first. Consider $\Z/p\Z$. Say I am allowed the two operations $x\mapsto x+1$ and $x\mapsto 2x$. Then, starting from $0$, I can ...
H A Helfgott's user avatar
  • 20.2k
27 votes
2 answers
2k views

Monstrous Moonshine for Thompson group $Th$?

I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2, $$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, ...
Tito Piezas III's user avatar
23 votes
2 answers
2k views

divisors of $p^4+1$ of the form $kp+1$

In group theory the number of Sylow $p$-subgroups of a finite group $G$, is of the form $kp+1$. So it is interesting to discuss about the divisors of this form. As I checked it seems that for an odd ...
BHZ's user avatar
  • 1,168
22 votes
1 answer
2k views

Monstrous moonshine for $M_{24}$ and K3?

An important piece of Monstrous moonshine is the j-function, $$j(\tau) = \frac{1}{q}+744+196884q+21493760q^2+\dots\tag{1}$$ In the paper "Umbral Moonshine" (2013), page 5, authors Cheng, Duncan, and ...
Tito Piezas III's user avatar
18 votes
2 answers
720 views

Infinitely many solutions to a particular embedding problem in Galois theory

Given a Galois extension of number fields $L/K$ and an exact sequence of groups $$1\to \ker \varphi\to G \overset{\varphi}{\to} \text{Gal}(L/K)\to 1$$ where $G$ is a finite group, $\ker \varphi$ is ...
user72870's user avatar
  • 181
18 votes
0 answers
2k views

$G$ a group, with $p$ a prime number, and $|G|=2^p-1$, is it abelian?

During my research I came across this question, I proposed it in the chat, but nobody could find a counterexample, so I allow myself to ask you : $G$ a group, with $p$ a prime number, and $|G|=2^p-1$, ...
Dattier's user avatar
  • 4,074
16 votes
1 answer
731 views

Transitive actions of finite subgroups of ${\rm GL}(n,\Bbb Z)$ on projective geometries

For any $n$, the group ${\rm GL}(n,\Bbb Z)$ has a natural action on $\Bbb Z^n$. Modding out a prime $p$ yields an action on the vector space $F_p^n$, where $F_p$ is the finite field with $p$ elements. ...
Joy Morris's user avatar
14 votes
5 answers
2k views

What is the Hilbert class field of a cyclotomic field?

In the answers to Qiaochu's post on defining representations of finite groups over the algebraic integers, it came out that which fields a representation of a finite group is defined over might depend ...
Ben Webster's user avatar
  • 44.7k
13 votes
4 answers
2k views

Which groups are Galois over some p-adic field?

Suppose I have some finite $p$-group $G$, or a little extension of it. How do I know if there exists a prime $l$ and a finite extension $K$ of $\mathbb{Q}_l$ such that $G$ is the Galois group of ...
Pablo's user avatar
  • 11.3k
13 votes
3 answers
475 views

Explicit construction of an element of ${\rm GL}(2, p)$ of order $p+1$

It is well-known that the order of $GL(2, p)$ is $(p^2-1)(p^2-p) = (p-1)^2(p+1)p.$ It is easy to construct matrices of orders $(p-1)$ and $p$ (diagonal and parabolic, respectively), but the only way ...
Igor Rivin's user avatar
  • 96.4k
13 votes
2 answers
668 views

On the sum of the subgroup orders of a finite group

Let $G$ be a finite group. Consider the function providing the sum of the subgroups orders $$\sigma(G) = \sum_{H \le G} |H|.$$ Note that if $C_n$ is cyclic of order $n$ then $\sigma(C_n) = \sigma(n)$, ...
Sebastien Palcoux's user avatar
12 votes
1 answer
450 views

abelian quotients of permutation groups

Let $G$ be a subgroup of the permutation group $S_n$, and let $H$ be a normal subgroup of $G$ such that the quotient group $G/H$ is abelian. What is the best known upper estimate for the cardinality $...
Yuri Bilu's user avatar
  • 1,294
12 votes
2 answers
876 views

Wedderburn's theorem for $\mathbb{Q}G$

Let $G$ be a finite group and let $\mathbb{Q}G=M_{n_1}(D_1)\times\cdots\times M_{n_k}(D_k)$ be the decomposition of $\mathbb{Q}G$ as a product of rings of matrices over divisions rings. Let $Z_i$ be ...
Diego Sulca's user avatar
12 votes
2 answers
893 views

Embeddings of finite groups into GL(n,Q_p)

This question is inspired by some interesting comments on this recent question. Fix an integer $n \geq 1$ and a finite subgroup $G$ of $\mathrm{GL}_n(\mathbf{C})$. It is known that there are ...
François Brunault's user avatar
12 votes
1 answer
455 views

Is there a classification of finite simple groups of perfect power order?

The finite simple group $\operatorname{PSp}(4,7)$ has order $138297600 = 11760^2$. There also seems to be a description of the $q$ such that $\operatorname{PSp}(4,q)$ has square order, see for example ...
spin's user avatar
  • 2,821
12 votes
4 answers
2k views

Mystery of the Monstrous Moonshine

There's a very famous group, the largest sporadic simple finite group, sometimes called a monster whose size is quoted below. What's the explanation that the primes appearing in it, ...
Ilya Nikokoshev's user avatar
11 votes
1 answer
416 views

Number of 1's in binary expansion of $a_n = \frac{2^{\varphi(3^n)}-1}{3^n}$

My question is about the Hamming Weight (or number of 1's in binary expansion) of $a_n = \frac{2^{\varphi(3^n)}-1}{3^n}$ A152007 For example, $a_3 = 9709 = (10110111101001)_2 $ has nine 1's in binary ...
Federico's user avatar
  • 113
11 votes
2 answers
577 views

About normal minimal subgroups not in the Frattini

In Neukirch--Schmidt--Wingberg, "Cohomology of Number Fields", Second edition, page 624, Exercise 2, it is stated the following fact. $\textbf{Claim}$: If $N$ is a normal subgroup, minimal among ...
Tom1909's user avatar
  • 113
11 votes
1 answer
364 views

Can we glue characteristic 0 and characteristic p representations of a finite group given equality of (Brauer) characters?

Suppose I have a prime $p$ and a finite group $G$ together with representations $\sigma: G \to GL_n(\mathbb{Q}_p)$ and $\pi: G \to GL_n(\mathbb{F}_p)$. My question is: When does there exist a ...
S. Carnahan's user avatar
  • 45.7k
11 votes
1 answer
685 views

On the order of finite simple groups

About the order of finite simple groups there exists a very interesting result which stated as follows: Let $G$ be an non-solvable simple group of order $g$. If $p\mid g$, where $p>g^{1\over 3}$ ...
BHZ's user avatar
  • 1,168
11 votes
2 answers
593 views

Characterization of finite groups using sum of the orders of their elements

Notation: If $G$ is a finite group, $o(g)$ denotes the order of the element $g\in G$. Motivation: Some finite groups could be uniquely determined by the size of the group. For example given a prime ...
Morteza Azad's user avatar
10 votes
4 answers
1k views

The powers of non-empty subset of a group that generate a subgroup

If G is a group and A and B to non-empty subsets of G, then by AB we mean the set consist of all product ab where a is in A and b is in B.(Standard definition) Similarly we can define X^m where X is a ...
katie's user avatar
  • 427
10 votes
1 answer
302 views

Finite subgroups of Lie group over algebraic ring of integers

I have frequently seen results like: There are 4 isomorphism types of finite subgroups of $SL_2(\mathbb{Z})$, namely $\mathbb{Z}_2,\mathbb{Z}_3,\mathbb{Z}_4,\mathbb{Z}_6$. I wonder what is known of ...
Simon Lentner's user avatar
9 votes
2 answers
794 views

Why do these finite group Dedekind matrices seem to have integer spectrum when specialized to the order of group elements?

Let $p$ be a prime and let $f_p$ be the permutation on the set $\{1,2,\cdots,p-1\}$ which is given by taking inverses in $\mathbb{Z}/(p)$: $$x \bmod(p) \mapsto \frac{1}{x} \bmod (p)$$ So for instance, ...
mathoverflowUser's user avatar
9 votes
0 answers
189 views

Cyclic numbers of the form $2^n + 1$

A cyclic number (or cyclic order) is a number $m$ such that the only group of order $m$ is the cyclic group $\mathbb{Z}/m\mathbb{Z}$. The set of cyclic numbers admits a couple of cute number-theoretic ...
Z. A. K.'s user avatar
  • 756
8 votes
2 answers
880 views

Moebius function of finite abelian groups

I am wondering if there is any literature on general formula of the Moebius function of subgroup lattices of any finite abelian group $G$? What I know is When $G$ is cyclic, the Moebius function is ...
JKDASF's user avatar
  • 231
8 votes
2 answers
479 views

Abundancy index and non-solvable finite groups

Let $\sigma$ be the sum-of-divisors function. A number $n$ is called abundant if $\sigma(n)>2n$. Note that the natural density of the abundant numbers is about $25 \%$. The abundancy index of $n$ ...
Sebastien Palcoux's user avatar
8 votes
1 answer
534 views

What is the minimum $k$ such that $A^k \equiv I$ mod p for invertible matrices?

Let $F$ be a finite field of order $p$, where $p$ is prime. For any $n\times n$ matrix $A$ that is invertible over $F$, then there would appear to exist integers $k$ such that $A^{k} = I$. My question ...
Joe Fitzsimons's user avatar
8 votes
1 answer
489 views

Elements living in the conjugacy class and in the centralizer of an $m$-cycle in $A_m$

Let $m>1$ be an odd natural number, $x$ a $m$-cycle in $A_m$, the alternating group in $m$ letters, $C$ the conjugacy class of $x$ in $A_m$. Question: How can I describe the elements in the set $\{ ...
Fernando's user avatar
7 votes
4 answers
768 views

How many finite simple groups of order $p+1$?

I'm looking at finite simple groups of order $p+1$ where $p$ is a prime number. But they don't seem to fall into any classification - have these all been determined? Is the number of them even ...
Dr Shello's user avatar
  • 1,180
7 votes
1 answer
332 views

Conjectured combinatorial non-equality

Let $n,k,\ell$ be integers for which $0\leq k<\ell \leq n-6$. For a fixed $n$, think of $k,\ell$ as being allowed to vary. I believe the values $$(n-k-5)(k+1)(k+2)\binom n{k+3}~~~\text{and}~~~(n-...
John McVey's user avatar
  • 1,068
7 votes
2 answers
1k views

How does one compute induced representations for modular representations?

The set-up is this: Let $G$ be a finite group, and $H$ a subgroup. We are given an irreducible representation of $H, \rho: H\rightarrow GL_n(K)$ (I will notationally identify $\rho$ with its character)...
Randy Brown's user avatar
  • 1,386
7 votes
1 answer
145 views

How to prove the relationship between Stern's diatomic series and Lucas sequence $U_n(x,1)$ over the field GF(2)?

I found the bit count of Lucas sequence $U_n(x,1)$ over the field GF(2) is Stern's diatomic series, I want to know the reason? https://oeis.org/A002487 : Stern's diatomic series https://oeis.org/...
phantom's user avatar
  • 317
6 votes
1 answer
2k views

Are there infinitely many insipid numbers?

A number $n$ is called insipid if the groups having a core-free maximal subgroup of index $n$ are exactly $A_n$ and $S_n$. There is an OEIS enter for these numbers: A102842. There are exactly $486$ ...
Sebastien Palcoux's user avatar
6 votes
2 answers
417 views

How simple does a $\mathbb{Q}$-simple group remain after base change to $\mathbb{Q}_{\ell}$?

Of course the general answer to the question in the title is: not very simple. I could not think of a better title, so let me explain my question in more detail. I have a number field $E/\mathbb{Q}$, ...
jmc's user avatar
  • 5,504
6 votes
1 answer
476 views

Structure of Deligne-Lusztig representations $R_{T,\theta}$ for ministropic $T$ and cuspidal representations

Let $G$ be a reductive group over a finite field $k$, let $F$ be a Frobenius morphism on $G$. I'll start with a somewhat vague question and make my question more specific further down: How do ...
John Binder's user avatar
  • 1,453
6 votes
1 answer
440 views

Applications of the Galois embedding problem

Given a finite Galois extension of number fields $L/K$ with Galois group $G$ and a surjection $E\twoheadrightarrow G$ of finite groups, the Galois embedding problem is the question of whether there ...
Bear's user avatar
  • 845
5 votes
1 answer
358 views

The number of polynomials on a finite group, II

This question is follow up of this MO-post. First let us recall the necessary definitions. A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\mathbb N$ and elements $a_0,...
Taras Banakh's user avatar
  • 41.8k
5 votes
1 answer
343 views

Large gaps in Singer planar difference sets?

By a classical result of Singer (1938), for a prime number $p$ the cyclic group $C_n$ of order $n=1+p+p^2$ contains a subset $D$ of cardinality $|D|=1+p$ such that $DD^{-1}=C_n$. Such set $D$ is ...
Taras Banakh's user avatar
  • 41.8k
5 votes
1 answer
555 views

Which cyclic groups admit a difference set?

Problem 1. For which $n$ does the cyclic group $C_n$ admit a difference set $D\subset C_n$, i.e., a set such that each non-unit element $x\in C_n$ can be uniquely written as the difference $x=ab^{-1}$...
Taras Banakh's user avatar
  • 41.8k
5 votes
2 answers
571 views

Exceptional isomorphisms between finite simple Chevalley groups

Steinberg's "Lectures on Chevalley Groups" https://math.depaul.edu/cdrupies/research/papers/chevalleygroups.pdf contain ``a complete list of isomorphisms" among the various finite simple Chevalley ...
Yuri Zarhin's user avatar
  • 5,050
5 votes
1 answer
360 views

Is the following variant of Shafarevich's theorem known?

Let $Q$ be a finite simple group which may be realized as the Galois group of some extension of $\mathbb{Q}$ (like for instance $PSL_2(\mathbb{F}_p)$ for $p\geq 5$, or the monster group) and let $G$ ...
user avatar
5 votes
0 answers
78 views

Conjugacy classes in normalized unit group of a group ring

Let $V(FA_4)$ be the normalized unit group of the group ring $FA_4$, where $F$ is the field containing 4 elements and $A_4$ is the alternating group on 4 symbols. How can I find conjugacy classes of ...
HIMANSHU's user avatar
  • 381
5 votes
1 answer
368 views

Boolean ring of unitary divisors / Structure of unitary divisors?

I hope this question is appropriate for MO: Let $n$ be a natural number, $U_n := \{ d | d \text{ divides } n, \gcd(d,n/d)=1\}$ be the set of unitary divisors. We can make $U_n$ to a boolean ring: $$a \...
user avatar