4
votes
2answers
104 views

How to prove $\frac{1}{q} \binom{p^k q}{p^k} \equiv 1 \mod p\;\;$?

An argument that I just came across asserts that if $p$ is prime, and $k, q\in \mathbb{Z}_{>0}$, then $$\frac{1}{q} \binom{p^k q}{p^k} \equiv 1 \mod p \;.$$ This assertion was made in a way (i.e. ...
2
votes
0answers
16 views

isometry group of an integer $n$ as of the corresponding $\Omega(n)$-parallelotope

Let $\prod_{i\in I}p_{i}^{a_{i}}$ be the prime factorization of a positive integer $n$ and let's consider the $\Omega(n)$-parallelotope built with, for all $i\in I$, $a_{i}$ pairwise orthogonal ...
2
votes
1answer
69 views

Are algebraic properties consistent among ALL types of number groups?

I'm embarking on a self study course of group theory, modular arithmetic, and other mathematics relating to cryptography. I notice that when studying modular arithmetic that they explicitly say that ...
2
votes
1answer
84 views

Are there groups with conjugacy classes as large as the divisors of perfect numbers?

Are there groups, where the conjugacy classes have elements according to the divisors of perfect numbers larger than $6$? I already got the first example, so therefore $6$ is excluded: $S_3$. The ...
2
votes
1answer
86 views

Cauchy-Davenport theorem and its extension

According to Cauchy-Davenport Theorem, if $A,B$ are subsets of a prime field ($F_p$) then we have the following bound on the number of elements within the sumset $A + B = \left\{ {\left. {a + b} ...
3
votes
1answer
88 views

The structure of $(\mathbb Z/525\mathbb Z)^\times$

I am working on the following problem. Find the number of the elements of order 4 in $(\mathbb Z/525\mathbb Z)^\times$. I tried to solve it in the following way: since $525=3\cdot5^2\cdot7$, we ...
4
votes
1answer
192 views

If $x,y,z \in \mathbb{Z}$ such that $x^4+y^4+z^4 \equiv 0 \pmod{29}$, prove that $x^4+y^4+z^4 \equiv 0 \pmod{29^4}$

If $x,y,z \in \mathbb{Z}$ such that $x^4+y^4+z^4 \equiv 0 \pmod{29}$, prove that $x^4+y^4+z^4 \equiv 0 \pmod{29^4}$. I have no idea where to start, but this is my abstract algebra homework, so I ...
0
votes
1answer
64 views

Diophantine equation and cyclicity of $\mathbb{F}_p^*$

I am trying to prove that the diophantine equation $$1998^2x^2+1997x+1995-1998x^{1998}=1998y^4+1993y^3-1991y^{1998}-2001y$$ has no solution in integers (given that $1997$ is a prime). To do so, ...
1
vote
1answer
209 views

Computing $\bmod$s with large exponents by paper and pencil using Fermat's Little Theorem.

I'm having a bit of trouble computing $\bmod{mod}$s of large numbers using Fermat's Little Theorem. For example, how would you compute $7^{435627650}\mod 13$? The solution given is $435627650\mod ...
5
votes
2answers
409 views

When is the group of units in $\mathbb{Z}_n$ cyclic?

Let $U_n$ denote the group of units in $\mathbb{Z}_n$ with multiplication modulo $n$. It is easy to show that this is a group. My question is how to characterize the $n$ for which it is cyclic. Since ...
5
votes
2answers
179 views

Upper bound for the sum of the orders in a finite group

This is a cute question that I found in ML. I thought a little bit about it and couldn't complete it to a solution. Let $G$ be a non-cyclic finite group of order $n.$ Let $S(G)=\sum_{g \in G} ...
0
votes
1answer
62 views

Prove if $g$ is an element of order $d$ and $d$ divides $n$ then $gn = 1$. [duplicate]

Prove if $g$ is an element of order $d$ and $d$ divides $n$, then $gn = 1$.
30
votes
2answers
443 views

Are there/Why aren't there any simple groups with orders like this?

The orders of the simple groups (ignoring the matrix groups for which the problem is solved) all seem to be a lot like this: ...
5
votes
5answers
375 views

Is $\mathbb Z _p^*=\{ 1, 2, 3, … , p-1 \}$ a cyclic group?

In undergraduate course, the two groups which are most frequently used may be $$\{ 0, 1, 2, ... , p-1\}$$ and $$\{ 1, 2, ... , p-1\}$$ where $p$ is a prime. The first one is a group under addition ...
4
votes
1answer
66 views

Abelian groups related to $(\mathbb{Z}/p\mathbb{Z})^*$, with order close to $p$ — analogous to elliptic curve groups, but simpler

I'm looking for a construction of a group $H$ that is "a sister" to $(\mathbb{Z}/p\mathbb{Z})^*$, in the following rough sense: Each element of $H$ can be represented by one or a few elements of ...
2
votes
1answer
59 views

a question on congruences

I would like to prove that the following 2 are equivalent: $\gcd(a,n)=1$ and $\exists x: x^m\equiv a \pmod n$ $a^{\frac{\phi(n)}{d}}\equiv 1 \pmod n$ where $d=\gcd(m,\phi(n))$ $\phi(n)$ is Euler ...
0
votes
1answer
83 views

How does one show that $(n-2)! = 2^{m-1} m! (n - 2m)!$ has only one solution for $n\ne 6$?

The obvious solution is 1, for $n=6$ there is another one - $m=3$. How does one show that for other $n$ there are no solutions but $m=1$? This is to show that for $n\ne 6$ all automorphisms of $S_n$ ...
0
votes
2answers
89 views

Finding $\mathbb{Z}_{p^{3}}(p)$

I have to find $\mathbb{Z}_{p^{3}}(p)$ (p is a prime). I know that the order of any element in $\mathbb{Z}_{p^{3}}$ divides $p^{3}$. How can I then deduce that ...
4
votes
1answer
327 views

Finding a generator of $(\mathbb Z/p\mathbb{Z})^*$

Is there a method for finding a primitive element (generator) of $(\mathbb Z/p\mathbb{Z})^*$, where $p$ is a prime number?
3
votes
1answer
64 views

Number counting functions related to simple groups and asymptotic law of distribution

We say that an positive integer $n$ is a simple number if there exist a non abelian simple group of order $n$. Denote by $\mathfrak{s}$ this set. prime-power number if it is of the form $n=p^a$, ...
1
vote
1answer
236 views

Fencing the Group size,and its implication to Finiteness of Tate-Shafarevich Group

This question is an interesting one,not like my previous one. Can we judge the size of a Quotient Group by seeing the size of its constituents ? To add something ,Suppose consider a group ...
2
votes
3answers
127 views

Finding a (small) prime great enough that there are at least m elements of order m

I'm hoping that someone can provide me with some results or point me in the right direction. I'm working with finite fields; really, I'm just doing arithmetic modulo a prime $p$. I'm taking elements ...
4
votes
8answers
524 views

Order of cyclic groups

Wikipedia says: It is known that $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic if and only if n is 1 or 2 or 4 or $p^k$ or $2p^k$ for an odd prime number p and k ≥ 1. The statement seems provable ...