3
$\begingroup$

Let $\mathbb{Z}[\frac{1}{p}]$ be the additive group of rational numbers of the form $mp^n$ where $m$, $n$ are elements of $\mathbb{Z}$ and $p$ is a fixed prime. Describe $\text{End}(\mathbb{Z}[\frac{1}{p})]$ and $\text{Aut}(\mathbb{Z}[\frac{1}{p}])$.

Unfortunately I have never done this type of exercise and do not know where to start: I have great difficulties in general because I do not know how to describe these types of groups. Can you help?

Sorry if my English is not exactly correct.

  • 2
    This may sound weird at first, but this wikipedia page should be helpful: http://en.wikipedia.org/wiki/Cauchy%27s_functional_equation#Proof_of_solution_over_rationals2011-05-19
  • 0
    Don't worry about your English -- it's not only better than in most other questions on this site; it's flawless (after someone correct a typo).2011-05-19
  • 0
    Myself's comment is actually incredibly helpful. It might also help to notice that Qp has a natural ring structure, that is, it is closed under multiplication.2011-05-19

2 Answers 2