1
vote
1answer
40 views

A direct proof of $\binom{m\,p^k-1}{p^k-1}\equiv1~(\text{mod $p$})$?

In Nathan Jacobson's "Basic algebra I" the exercises 1.13.11-14 prove the following extension of (a part of) the Sylow's second theorem: If $p$ is a prime and $p^k\bigm||G|$, then the number of ...
2
votes
1answer
49 views

About a nested sequence of subsets of integers

Let $(H_n)$ be a sequence of nonempty subsets of $\Bbb Z$ such that always $\{a-b\mid a,b\in H_{n+1}\}\subsetneqq H_n$. Can we deduce that there is some $n$ such that $\{a-b\mid a,b\in H_{n}\} = ...
2
votes
2answers
88 views

$\forall a \in \mathbb{Z}, \quad a^{13} = a \bmod 35$

How does one prove that $\forall a \in \mathbb{Z}, \quad a^{13} = a \bmod 35$? I would recall that $a^p = a \bmod p$ if $p$ does not divide $a$, but $13 \neq 35$ and besides the statement should ...
5
votes
3answers
80 views

Why $f\colon \mathbb{Z}_n^\times \to \mathbb{Z}_m^\times$ is surjective?

If $m|n$. Why the map $f\colon \mathbb{Z}_n^\times \to \mathbb{Z}_m^\times$ given by $a \mod{n}\mapsto a \mod m$ is a surjective homomorphism of groups? Attempt: I proved it is well a well defined ...
4
votes
2answers
48 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
221 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
98 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
75 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
63 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
29 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
111 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
134 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
34 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
56 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
51 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
70 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
286 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
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. ...
1
vote
1answer
118 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
45 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
144 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
46 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
99 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
91 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
135 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
39 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
57 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
44 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
167 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
73 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
73 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
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 ...
9
votes
3answers
309 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