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
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.