4
votes
2answers
46 views

Density in $\mathbb{R}_{ +}$ of a subgroup of $\mathbb{Q}_{> 0}$?

Let $\phi : \mathbb{Q}_{>0} \to \mathbb{Z}$ be the group morphism defined by $\phi(p) = p$ for $p$ a prime number. It follows that $\phi(1)=0$, $\phi(a.b) = \phi(a)+\phi(b)$, $\phi(a^{-1}) = ...
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, $H = \langle 2 \rangle$). ...
1
vote
1answer
39 views

Order of some $\alpha$ in $S_4$

I want to work out the order of $\alpha = \left[ \begin{align} & 1&2&&3&&4&\\&4&2&&1&&3& \end{align} \right]$ in $S_4$ Now when I think of ...
4
votes
4answers
219 views

How to show a group is cyclic?

One question asking if $\mathbb{Z}^*_{21}$ is cyclic. I know that the cyclic group must have a generator which can generate all of the elements within the group. But does this kind of question ...
0
votes
1answer
26 views

$U(\mathbb{Z}/n\mathbb{Z})$ Question

Consider the group $U(\mathbb{Z}/n\mathbb{Z})$. If $e$ is the order of the group, then $\forall a\in U(\mathbb{Z}/n\mathbb{Z})$, $a^e\equiv 1\pmod{n}$. But why is this true? Why is it that there ...
3
votes
2answers
96 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 ...
2
votes
1answer
74 views

An exact sequence of unit groups

In the answer of K. Conrad to this question, he mentions a "nice 4-term short exact sequence of abelian groups (involving units groups mod a, mod b, and mod ab)" proving the product formula for ...
0
votes
1answer
46 views

How would one find a) All the primitive characters modulo 8, b) All the non-primitive characters modulo 8?

Preferably explained in novice terms! I can start it off by having the multiplicative group modulo 8 with elements $[1], [3], [5], [7]$ and not sure where to go now. I see there is a similar question ...
1
vote
1answer
35 views

What can we say about this quantity?

Let $\phi(n)$ be the Euler phi-function. If $a>1$ is an integer, then what is the remainder when $\phi(a^n - 1)$ is divided by $n$ in accordance with the Euclidean algorithm?
1
vote
3answers
62 views

Calculating $x^{103} \equiv 2 \pmod{143}$

I need to find x, given that: $0\leq x \leq 143$ and $x^{103}\equiv 2 \pmod{143}$. I tried to use Euler's theorem $p(143)=120$, but it didn't help.
2
votes
0answers
27 views

Which integers are a product of partition numbers?

What can be said about the set of positive integers representable as a product of the form $p(n_1)\cdots p(n_r)$ for the partition function $p(n)$ ? Such numbers $k$ arise as the number of distinct ...
2
votes
2answers
31 views

Show that if $p$ is a prime of the form $p=4n+1$, then we can solve $x^2\equiv -1\mod p$(with $x$ an integer).

Show that if $p$ is a prime of the form $p=4n+1$, then we can solve $x^2\equiv -1\mod p$(with $x$ an integer). My attempt:If $p$ is a prime, then $U_p=${$[x]|1\leq x<p$} is cyclic.
0
votes
2answers
108 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
65 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
133 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
1answer
31 views

Is $GL_2(\mathbb{Z}_p)$ a maximal compact subgroup of $GL_2(\mathbb{Q}_p)$?

Let $\mathbb{Q}_p$ be the set of all p-adic numbers and $\mathbb{Z}_p$ the set of all p-adic integers. Is $GL_2(\mathbb{Z}_p)$ a maximal compact subgroup of $GL_2(\mathbb{Q}_p)$? Thank you very ...
0
votes
0answers
26 views

Closure of divisibility in denumerator, under sum of fractions

I have to prove that for a fixed positive integer n, the subset A of Q consisting of rationals with denumerator that divide n under addition, forms a group under addition. I just did that it's ...
0
votes
1answer
55 views

Functions determined by characters are linearly independent?

Let $X$ be a set with an action on $\mathbb{Z}/N\mathbb{Z}$. For a Dirichlet character $\chi \pmod N$ we set $$R(\chi)=\left\{ f:X \to \mathbb{C} ~\mid~ f(l s)=\chi(l)f(s) \text{ for all } l\in ...
3
votes
1answer
50 views

Does every finite cyclic group appear as a subgroup of the multiplicative group of a finite field?

Does every finite cyclic group appear as a subgroup of the multiplicative group of a finite field? In other words, given any $d \in \mathbb{N}$, can we find a prime $p$ and $k \in \mathbb{N}$ such ...
2
votes
1answer
69 views

Prove that S consists of the quadratic residues mod (p) and T consists of the quadratic non-residues mod (p)

All the followings are $\bmod$ $p$ Let p be an odd prime. Suppose that the set $X = \{ 1,2, . . . , p-1\}$ can be written as the union of two nonempty subsets $S$ and $T$, where $S \neq T$, such that ...
0
votes
3answers
219 views

How many elements are in the invertible set Zn?

My question is directly, how many elements are in the invertible set Z35? It's my understanding that for any Zn, if n is prime, then the number of invertible elements is equal to n-1. In addition, ...
0
votes
0answers
50 views

show $x^p−x\equiv x(x−1)(x−2)⋯(x−(p−1)) \pmod p$

I have been trying a long time figure out how to state the following congruent show $x^p−x\equiv x(x−1)(x−2)⋯(x−(p−1)) \pmod p$ I got some hint: find all the roots of $x^p − x$ using Fermat's ...
0
votes
2answers
46 views

How to determine if some $x$ is a generator of a subgroup of $Z^{*}_{y}$ of order $a$

Suppose we have integers $x,y$ and the prime factorization of $y-1$, and further suppose that $a$ is the largest prime factor of $y-1$ and that $y$ is prime. How do you determine if $x$ is a generator ...
4
votes
2answers
110 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. ...
1
vote
1answer
112 views

Find period of power sequence $a^k \mod m$, with $a, m$ not coprime

Let $a, m$ be positive integers and $m > 1$. I'm interested in the sequence $(a^k)_{k \in \mathbb{N_0}} \mod m$. Since there are only $m$ different values that can occur in the sequence and since ...
1
vote
0answers
44 views

Divisors of the totient function and congruences

Based on the question Question about the totient function and congruence classes, I would like to ask two new questions: Question 1 Does it hold that if $k\mid\varphi(n)$ then the elements of the ...
1
vote
2answers
137 views

Does someone know why raising the element of a group to the power of the order of the group yields the identity?

Does someone know the why raising the element of a group to the power of the order of the group yields the identity? By (finite) group I mean a tuple (G,*) that satisfies the following: closure ...
1
vote
0answers
45 views

Decomposition subgroup of Cyclic Galois Extension

My question: Say we have a cyclic Galois extension of degree $n$ over $\mathbb{Q}$. Denote the Galois group as $G$. If $H\leq G$, then does there exist a prime, $q$ in $\mathbb{Z} \subset \mathbb{Q}$ ...
3
votes
2answers
96 views

2 is a primitive root mod $3^h$ for any positive integer $h$

It's easy to verify that 2 is a primitive root mod $3^2$. But then why does it follow that 2 is a primitive root mod $3^h$ for any positive integer $h$? This was used in the solution of 2009 Putnam ...
4
votes
0answers
86 views

Role of the discrete subgroups of Lie groups

This is a question I don't believe is too vague to admit a sensible answer: In what directions can the structure theory of the discrete subgroups of real and p-adic Lie groups applied? What ...
2
votes
1answer
134 views

Subgroup and index in $\mathbb{Z}^2$

Which of the following are finite index subgroups in $\mathbb{Z}^2$? What is the index?? $H=\{(x,y) \in \mathbb{Z}^2 \text{ s.t. } x+y=1\}$ $H=\{(x,y) \in \mathbb{Z}^2 \text{ s.t. } x+y=0\}$ ...
0
votes
1answer
43 views

Is 2 a generator of $(\mathbb Z/5^k\mathbb Z)^\times$

If not, is there a $k\ge 1000$, for which it is? I want to know this, because it will then fit in my proof that there exists a power of $2$ whose last $1000$ digits are either $1$ or $2$.
2
votes
1answer
37 views

Integer subgroup of indefinite orthogonal group

I'm interested in the subgroup of the indefinite orthogonal group consisting of integer matrices i.e. $$ZO(p,q):=\{M \in GL(p+q, \mathbb{Z})| M^TI_{p,q}M=I_{p,q}\},$$ where $I_{p,q}$ denotes the ...
0
votes
1answer
56 views

Kernel of a homomorphism is subgroup of squares

Let $\gamma:(\mathbb{Z}/p^m\mathbb{Z})^*\rightarrow \{1,-1\}$ be defined as $\gamma(a)=(\frac{a}{p})$, the Legendre symbol; $p$ is an odd prime, and $m$ is an integer greater or equal to $1$. I have ...
5
votes
4answers
118 views

${{p-1}\choose{j}}\equiv(-1)^j \pmod p$ for prime $p$

Can anyone share a link to proof of this? $${{p-1}\choose{j}}\equiv(-1)^j(\text{mod}\ p)$$ for prime $p$.
1
vote
1answer
28 views

Are these conditions enough to specify a unique number? (counting sylow p-subgroups)

Sylow's third theorem gives the two facts that the number of sylow p-subgroups $n_p$ of a group $G$, whose order we can write as $|G| = p^rm$ such that $p\not |\ m$ will satisfy both $n_p | m$ and ...
1
vote
1answer
43 views

Order of modular group

Prove $|(\mathbb{Z} / p^e \mathbb{Z} )^{\times}| = p^e - p^{e-1}$ I know it has something to do with the fact that we have $p^e$ elements and we're substracting $p^{e-1}$ multiples of $p$, but I'd ...
3
votes
1answer
101 views

Orbits of $\mathbb{Z}_n^{*}$ acting on a set $\mathbb{Z}_n$

Let $n\geq 2$ be an integer and consider the action $\Phi: \mathbb{Z}_n^{*}\times \mathbb{Z}_n \rightarrow \mathbb{Z}_n$ defined as $$\Phi(\alpha)(x)=(\alpha x \textrm{ mod } n),$$ i. e. simply the ...
3
votes
1answer
127 views

Finite abelian unramified $p$-extension of a number field

Let $K$ be a number field. How many finite abelian unramified $p$-extensions of $K$ are there and what are their Galois groups? My feeling is, that every group $\mathbb{Z} / p^n \mathbb{Z}$ can occur ...
2
votes
1answer
72 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 ...
4
votes
1answer
105 views

Euler function and $\mathbb{Z}/n\mathbb{Z}$

I am trying to solve a very interesting problem about the ring $\mathbb{Z}/n\mathbb{Z}$ and Euler function $\phi (n)$, but i am not sure how to start, i have a few ideas, but none of them leads me to ...
2
votes
1answer
110 views

Given $G$ is a group and $a,b\in G $ and $ab=ba$. Prove…

$ab^n= b^na\;\; \forall n \in \mathbb{Z}$ I have been able to prove this for $n=0$ and for a positive integer (using induction). But for $n$a negative integer, I'm not able to prove it: $n=-m$ for ...
2
votes
1answer
72 views

A problem about the discrete logarithm

suppose there are a multiplicative cyclic group $F_p^*(p \;is\;big\; prime)$, and $G=\langle g \rangle(g \;is\; a\; generator)$ is a subgroup of it and $G$'s order is $q(q\;is\;big\;prime \;and ...
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 ...
6
votes
3answers
278 views

Additive group of rationals has no minimal generating set

In a comment to Arturo Magidin's answer to this question, Jack Schmidt says that the additive group of the rationals has no minimal generating set. Why does $(\mathbb{Q},+)$ have no minimal ...
-1
votes
3answers
138 views

Are the irrationals + zero an additive group?

Is it true that the only way two irrationals can sum to a rational is if they sum to zero? Thanks! -Dan
1
vote
1answer
65 views

Is this group isomorphic to $\mathbb{Z}_k$? The integers with multiples of $k$ subtracted until you're under $r$.

Where as the elements of the ring of integers modulo $n$ can be found by taking the integers and subtracting multiples of $n$ until you're just under $n$, consider the structure formed by taking ...
2
votes
1answer
46 views

Proof of $a^{2^p}=x^pax^{-p}=a$ $\forall a \in G$

I have to complete this exercise: "Let $G$ be a group with an element of finite order $n>1$ and exactly two conjugacy classes. Prove that $|G|=2$" The author gives some hints: " Prove the ...
2
votes
0answers
48 views

Abelian SubGroup Variant:

Consider the following problem: Find integers $x_1, x_2, x_3,\dots, x_n$ Such that: $$P(x_1,x_2,\dots, x_n) = Q$$ for some integer $Q$ and polynomial $P$ where for all permutations of any set of ...
3
votes
1answer
95 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 ...