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Isomorphism between groups

  1. Jun 9, 2012 #1
    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
  2. jcsd
  3. Jun 9, 2012 #2
    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.
  4. Jun 9, 2012 #3
    thanks, i finally have found that it is not true generally
  5. Jun 9, 2012 #4
    did you find the counterexample?
  6. Jun 9, 2012 #5


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    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.

    Last edited: Jun 9, 2012
  7. Jun 9, 2012 #6

    Ah, the above reminded me: the free group [itex]\,F_2\,[/itex] 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 [itex]\,(F_2)'=[F_2:F_2]\cong F_\infty\,[/itex]) , so we have injections [tex]F_2\to F_\infty\,\,,\,\,F_\infty\to F_2[/tex] but the two groups are clearly non-isomorphic.

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