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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

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 Paul Posts: 764 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

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?
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>
>
> Assuming that the generators for G are a subset of the generators
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> for H, so that H _is_ an extension of G:
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>
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> I've seen it said
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>
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> http://en.wikipedia.org/wiki/Free_group#Universal_property
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>
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> 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

Date Subject Author
7/9/13 David C. Ullrich
7/9/13 Sandy
7/14/13 Paul
7/14/13 Sandy
7/14/13 Paul
7/15/13 Butch Malahide
7/15/13 Butch Malahide
7/15/13 Sandy
7/15/13 David C. Ullrich
7/14/13 Butch Malahide
7/14/13 Sandy