0
votes
4answers
50 views

Show that some group is isomorphic to $\mathbb{Z_n}$

If $G$ has order $4$ and has an element of order $4$, then $G$ is isomorphic to $\mathbb{Z_4}$. Can someone briefly explain why this is true? I understand that $|G| = 4$, but I don't understand ...
0
votes
2answers
44 views

Sum of the elements of a cyclic subgroup

Let $G$ be a finite cyclic subgroup of the group of units of a commutative unital ring $R$. What is the sum of the elements of $G$, i.e. $$\sum_{x \in G}{x}?$$ [The answer is not difficult in the ...
3
votes
1answer
57 views

Hecke Operators

I'm currently working my way through the section on Hecke operators in Serre's book. In the proof that $ T(m)T(n) = T(mn) $ for all $ m,n \in \mathbb{Z}_{\geq 1} $ with $ \textit{gcd}(m,n) = 1 $. In ...
3
votes
1answer
57 views

An elementary question regarding a multiplicative character over finite fields

Reading Chapter 2 of Koblitz's Introduction to Elliptic Curves and Modular Forms, I got stuck on the following question. I would like to proceed my reading, so I would appreciate any hint to this. I ...
2
votes
1answer
63 views

Density of nilpotent numbers

A natural number $n$ is nilpotent if every group of order $n$ is nilpotent (equivalently, a direct product of Sylow subgroups). A natural number $n$ has nilpotent factorization if $\ell\not\equiv1$ ...
1
vote
1answer
30 views

Understanding the elements in groups(modulo, cyclic and other(?))

Question exactly as given on past exam: Consider the group $G=(\mathbb{Z}/15\mathbb{Z})^\times$ (under multiplication). Let $H$ be the subgroup generated by $2$ (that is, $H = \langle 2 \rangle$). ...
3
votes
2answers
106 views

Is there any usage of this sum formula?

Let $\phi$ be the Euler phi function, then $$n=\sum_{d|n}\phi(d)$$ It is one of the most famous formulas in number theory. It can be generalized by using groups. Let $G$ be a group of order $n$ and ...
0
votes
2answers
113 views

$\psi (m)\leq \phi (m)$ or $\psi (m) \geq \phi (m)$ when $\psi (m)\neq 0$?

(This is different than If $x^m=e$ has at most $m$ solutions for any $m\in \mathbb{N}$, then $G$ is cyclic) I was trying to solve this: Let $G$ be a finite abelian group of order $n$ for which the ...
1
vote
4answers
66 views

If p is a prime number of the form $4n+3$, show that we cannot solve $x^2\equiv -1\mod p$

Hint: Use Fermat's Theorem that $a^{p-1}\equiv 1\mod p$ if $p \nmid a$. (I have no idea, but something in group theory should help)
1
vote
2answers
143 views

The Proof of Wilson's Theorem using the auxiliary multiplicative modulous group [duplicate]

(self answered question, thanks for the hints Derek Holt provided:-)) problem 18,section 4 chapter 2 in Herstein's abstract algebra: Using the results of Problem 15 and 16,prove that if p is an ...
4
votes
2answers
112 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
19 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
76 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
85 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
100 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
98 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
193 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
65 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
279 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
678 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
200 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
64 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 $gn = 1$.
31
votes
2answers
462 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
474 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
71 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
62 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
360 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
69 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
245 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
130 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
539 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 ...