Can a group have monomorphisms in both directions and still not be isomorphic?

[charlamov]

Is it true that if there is monomorphism from group A to group B and monomorphism from group B to group A than A and B are isomorphic? i need some explanation. thx

 

[algebrat]

Yes, I think that is true. At least for finite groups, because an injection both ways implies they have the same size, which means it is a bijection. Not sure about infinite groups, a good strategy for trying to find if something is true or not, is try proving either and see what facts you lack, try to construct a counterexample, which may help continue trying to prove truth, back and forth till you realize if it's true or not.

So if you can't see a proof for the infinite case as I haven't, try constructing a counterexample.

 

[charlamov]

thanks, i finally have found that it is not true generally

 

[algebrat]

did you find the counterexample?

 

[mathwonk]

i guess the usual example is of free (non abelian) groups on different sets of generators.

as i recall, a figure eight knot has a fundamental group which is free on two generators. Then you construct a covering space which is a wedge of more than two loops.

The covering space has fundamental group free on more generators, but injects into the fundamental group of the base.

Just recalling a lecture by Eilenberg from 30-40 years ago. I'll check it out.

here's a reference in hatcher's free algebraic topology book, pages 57-61.

http://www.math.cornell.edu/~hatcher/AT/ATpage.html

 

[DonAntonio]

i guess the usual example is of free (non abelian) groups on different sets of generators.

as i recall, a figure eight knot has a fundamental group which is free on two generators. Then you construct a covering space which is a wedge of more than two loops.

The covering space has fundamental group free on more generators, but injects into the fundamental group of the base.

Just recalling a lecture by Eilenberg from 30-40 years ago. I'll check it out.

here's a reference in hatcher's free algebraic topology book, pages 57-61.

http://www.math.cornell.edu/~hatcher/AT/ATpage.html

Ah, the above reminded me: the free group $\,F_2\,$ on two generators contains as a subgroup the free group on any number of generators up to and including the free group on infinite countable generators (for example, the group's commutator subgroup $\,(F_2)'=[F_2:F_2]\cong F_\infty\,$) , so we have injections $$F_2\to F_\infty\,\,,\,\,F_\infty\to F_2$$ but the two groups are clearly non-isomorphic.

DonAntonio