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 <email@example.com> > > 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.