# Tagged Questions

**2**

votes

**1**answer

206 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 ...

**21**

votes

**0**answers

640 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)\, ...

**20**

votes

**1**answer

792 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 ...

**4**

votes

**1**answer

425 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 ...

**4**

votes

**1**answer

305 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 ...

**10**

votes

**2**answers

547 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 ...

**12**

votes

**2**answers

587 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 ...

**8**

votes

**3**answers

531 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 ...

**5**

votes

**0**answers

250 views

### Vanishing sums of powers modulo n

Is it possible to describe all positive integer sequences $\{x_i\}$ such that
$$
\sum_i x_i=n
\quad
\hbox{and}
\quad
\sum_i x_i^k\equiv 0\pmod n
\quad
\hbox{for all $k$ (and a given $n$)?}
$$
...

**0**

votes

**2**answers

423 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 ...

**41**

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 ...

**6**

votes

**2**answers

471 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 ...

**37**

votes

**2**answers

1k 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|, ...

**6**

votes

**4**answers

605 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 ...

**7**

votes

**2**answers

383 views

### Elements living in the conjugacy class and in the centralizer of an m-cycle in Am

Let m>1 be an odd natural number, x a m-cycle in Am, the alternating group in m letters, C the conjugacy class of x in Am.
Questiom: How can I describe the elements in the set { j | x^j in C} in ...

**10**

votes

**4**answers

1k 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,
...

**11**

votes

**5**answers

1k 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 ...