Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math.independent

Topic: Free group on m generators elementary extension of the free group on n generators (n < m)?
Replies: 11   Last Post: Jul 15, 2013 1:13 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Paul

Posts: 407
Registered: 7/12/10
Re: Free group on m generators elementary extension of the free group
on n generators (n < m)?

Posted: Jul 14, 2013 5:08 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On Tuesday, July 9, 2013 4:40:34 PM UTC+1, dull...@sprynet.com wrote:
> On Tue, 09 Jul 2013 11:39:00 +0100, Sandy <sandy@hotmail.invalid>
>
> wrote:
>
>
>

> >For n, m natural numbers, n < m, let G be the free group on n generators
>
> >and H the free group on m generators. Is H an elementary extension of G?
>
>
>
> Assuming that the generators for G are a subset of the generators
>
> for H, so that H _is_ an extension of G:
>
>
>
> I've seen it said
>
>
>
> http://en.wikipedia.org/wiki/Free_group#Universal_property
>
>
>
> that any two free groups have the same first-order theory...


Either it's trivially true that whenever G is a subgroup of H, H is an elementary extension of G, or I'm confused about what "elementary extension" means. If I'm not confused here, then what I said is a trivial fact which would solve the OP's question. [I consulted the wiki and quickly read the reference to model theory].

Assume I'm confused. Could you give an example of G being a subgroup of H but H not being an elementary extension of G?

If I'm not confused then I don't think it matters (contrary to what Fred says) that the OP didn't say that n > 1.

Paul Epstein



Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.