Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Can anyone give me a reference to the aforementioned theorem? W. Hodges uses it for an example in his "Model Theory", but I couldn't find anything on it yet.

The group may be (let's say, countably) infinite, the direct sum will then be infinite, too, of course. The "exponent" of a group is defined here as $\sup_{g \in G} \mathopen|G.g\mathclose|$

share|cite|improve this question
I think this is usually called Prufer's first theorem (I don't know how to do the umlaut), and can be found in any group theory book which does a fair amount on abelian groups: Kaplansky's book, for example. – user641 Jul 23 '11 at 20:16
Solved by Steve D's comment above. Thanks a lot! (can't mark this question solved as I wasn't logged in when I asked it) – rainmaker Jul 24 '11 at 10:22
The moderators have merged your two accounts. If @Steve would post his answer as an answer, you can accept it. Otherwise, if would also be nice if you give a some-what detailed account of the proof below yourself and accept that as the answer. Cheers. – Willie Wong Jul 24 '11 at 13:41
up vote 5 down vote accepted

It's called Prüfer's first theorem and I found it on page 173 in

A. G. Kurosh: The Theory of Groups, Volume One, Second English Edition. Chelsea Publishing Company, New York 1960

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.