0
votes
0answers
27 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
109 views

Need help with proving that group is not finitely-generated [duplicate]

I need to prove that $(\mathbb{Q}^*, \times)$ (i.e rationals, zero excluded, under multiplication) is not finitely generated. So, suppose that G is finitely-generated. That means there exist a ...
1
vote
1answer
103 views

An example of a group operation on the rationals, which is not isomorphic to the additive group

I'm looking for an example of a group operation on the rationals, which is not isomorphic to the rational additive group. Can you find such an example?
2
votes
4answers
354 views

Show that the integers have infinite index in the additive group of rational numbers.

Show that the integers have infinite index in the additive group of rational numbers. Anybody in a good enough mood to tell me how this is done?
9
votes
4answers
392 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 ...
-2
votes
1answer
93 views

set of terminating decimals

Let $T\subset\mathbb Q$ be the set of all positive rational numbers that can be represented by a terminating decimal (in base 10), that is, a decimal whose tail consists of an infinite sequence of ...
6
votes
2answers
249 views

For what algebraic curves do rational points form a group?

For what real algebraic curves do rational points form a group ? How does this relate to Jacobian Varieties ?
1
vote
1answer
163 views

Properties of homomorphisms of the additive group of rationals

Let $f : (\mathbb{Q},+) \longrightarrow (\mathbb{Q},+)$ be a non-zero homomorphism. Can we conclude that $f$ is bijective (or, if that fails, that $f$ is injective or surjective)? Context The ...
7
votes
2answers
958 views

How can we find and categorize the subgroups of R?

$\newcommand{\R}{\Bbb R}\newcommand{\Q}{\Bbb Q}\newcommand{\Z}{\Bbb Z}$ What are all the subgroups of R = $(\R, +)$ and how can we categorize them? I started thinking about this question last night ...
17
votes
4answers
466 views

What do the cosets of $\mathbb{R} / \mathbb{Q}$ look like?

$\newcommand{\R}{\Bbb R}\newcommand{\Q}{\Bbb Q}$ Looking at the group of real numbers under addition $(\R, +)$ it contains the (normal) subgroup of rational numbers $(\Q, +)$. I am wondering how to ...
2
votes
1answer
141 views

Why is the group of rational numbers with odd denominators residually finite?

I want to prove that the additive group of rational numbers with odd denominators is residually finite. However, despite pondering for quite some time, I can't even find a single subgroup of finite ...