All Questions
Tagged with finite-groups nt.number-theory
88 questions
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 ...
- 41.8k
16
votes
1
answer
730
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. ...
- 163
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 ...
- 181
1
vote
0
answers
194
views
The number of fixed points of an automorphism of $\mathbb{Z}_m\times\mathbb{Z}_n$
Let $m$ and $n$ be two positive integers such that the groups $\mathbb{Z}_m$ and $\mathbb{Z}_n$ have no common direct factor. Then an automorphism $f$ of $\mathbb{Z}_m\times\mathbb{Z}_n$ is of type
$$\...
5
votes
2
answers
571
views
Exceptional isomorphisms between finite simple Chevalley groups
Steinberg's "Lectures on Chevalley Groups"
https://math.depaul.edu/cdrupies/research/papers/chevalleygroups.pdf
contain ``a complete list of isomorphisms" among the various finite simple Chevalley ...
- 5,050
32
votes
3
answers
3k
views
Is there a nice explanation for this curious fact about cyclic subgroups?
Here's something that I noticed that quite surprised me.
Let $G$ be a finite abelian group. Consider the following expression.
$$
\nu(G) = \sum_{\substack{H \leq G \\ H \text{ is cyclic}}} |H|
$$
It ...
- 6,290
13
votes
4
answers
2k
views
Which groups are Galois over some p-adic field?
Suppose I have some finite $p$-group $G$, or a little extension of it.
How do I know if there exists a prime $l$ and a finite extension $K$ of $\mathbb{Q}_l$ such that $G$ is the Galois group of ...
- 11.3k
6
votes
1
answer
440
views
Applications of the Galois embedding problem
Given a finite Galois extension of number fields $L/K$ with Galois group $G$ and a surjection $E\twoheadrightarrow G$ of finite groups, the Galois embedding problem is the question of whether there ...
- 845
13
votes
3
answers
475
views
Explicit construction of an element of ${\rm GL}(2, p)$ of order $p+1$
It is well-known that the order of $GL(2, p)$ is $(p^2-1)(p^2-p) = (p-1)^2(p+1)p.$
It is easy to construct matrices of orders $(p-1)$ and $p$ (diagonal and parabolic, respectively), but the only way ...
- 96.4k
11
votes
2
answers
593
views
Characterization of finite groups using sum of the orders of their elements
Notation: If $G$ is a finite group, $o(g)$ denotes the order of the element $g\in G$.
Motivation: Some finite groups could be uniquely determined by the size of the group. For example given a prime ...
6
votes
1
answer
476
views
Structure of Deligne-Lusztig representations $R_{T,\theta}$ for ministropic $T$ and cuspidal representations
Let $G$ be a reductive group over a finite field $k$, let $F$ be a Frobenius morphism on $G$.
I'll start with a somewhat vague question and make my question more specific further down:
How do ...
- 1,453
2
votes
1
answer
211
views
Finite sequences which happen to be the sequence of orders of elements of a simple group
Let $n\in \Bbb N$. Any finite group $G$ with $|G|=n$ has a solution for the Diophantine equation
$$\sum_{d|n}x_d\phi(d)=n~~~~~~~~~~~~~~~~~~~~~~~(1)$$
where $\phi$ is the Euler's totient function, $d$ ...
- 1,145
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}$, ...
- 5,504
2
votes
0
answers
235
views
Does knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ sometimes offer a significant advantage in finding $r$?
Is there a cyclic group $\mathcal G$ with generator $g$ for which the discrete log problem is assumed to be hard, but knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ for random $r$ makes finding $r$ easy?
...
- 4,943
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 ...
- 1,168
3
votes
0
answers
102
views
Localized at $p$ integral representations of finite elementary $p$-groups
Let $C_p$ be a cyclic group of prime order $p$.
Let $F=C_p^n=C_p\times\dots\times C_p$ ($n$ times).
I would like to to classify finite dimensional representations of $F$ over ${\mathbb{Z}}$.
However, ...
- 14.1k
3
votes
1
answer
608
views
Representation of GL(n, F_p) over F_p, for n small
The question is related to this post
Representation theory of the general linear group over a finite prime field
However, I am asking for more detailed references for n small, for example, for n=2, ...
- 31
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}$ ...
- 1,168
4
votes
1
answer
328
views
Maximum length of chains of subgroup in GL(n,q)
Let G=GL(n,q) be a general linear group n-dimensional over a field with q element (q power of a prime). I am looking for an estimate of maximum length of chains of subgroup in G.
Thanks.
- 41
2
votes
1
answer
229
views
on the prime divisors of $(p^2+1)/2 $
The following question is equivalent to a problem in group theory.
Let $ p > 13$ be a prime number distinct from 239. Let $ a=(p^2+1)/2 $. Is there any prime divisor $r$ of $a$ such that $r\mid ...
- 1,168
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)\, ...
- 12.6k
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 ...
- 12.6k
4
votes
1
answer
475
views
What natural numbers can be considered as the product of orders of elements of a finite (abelian) group
Problem A5 from putname's competition 2009 asks to prove that there is no finite abelian group such that the product of order of its elements is equal to $2^{2009}$. Starting from this problem, for ...
user30230
4
votes
1
answer
458
views
Does there exist an order in a number field of deg>1 with a map to F_p for all p?
This question is motivated by a computational issue. Suppose $R$ is a product of orders in numberfields such that there is no ring homomorphism $R \to \mathbb Z$, then can one write an algorithm that ...
- 2,224
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 ...
- 20.8k
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 ...
- 657
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 ...
- 427
5
votes
0
answers
294
views
Vanishing sums of powers modulo n
Is it possible to describe all positive integer sequences $\{x_i\}$ such that
$$
\sum_i x_i=n
\quad
\hbox{and}
\quad
\sum_i x_i^k\equiv 0\pmod n
\quad
\hbox{for all $k$ (and a given $n$)?}
$$
...
- 3,932
1
vote
3
answers
1k
views
primes dividing binomial coefficients
Dear All,
I am considering maximal subgroups of odd index in Alternating and Symmetric groups, and this leeds me to some questions on binomial coefficients that I presently do not know and that I ...
- 195
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 ...
- 65.4k
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)...
- 1,386
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|, ...
- 27.7k
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 ...
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 ...
- 1,180
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 ...
- 20.2k
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 $\{ ...
- 81
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,
...
- 15.1k
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 ...
- 44.7k