Isomorphism between groups

  • Thread starter charlamov
  • Start date
  • #1
11
0
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
 

Answers and Replies

  • #2
428
1
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.
 
  • #3
11
0
thanks, i finally have found that it is not true generally
 
  • #4
428
1
did you find the counterexample?
 
  • #5
mathwonk
Science Advisor
Homework Helper
2020 Award
11,206
1,414
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
 
Last edited:
  • #6
606
1
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 [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.

DonAntonio
 

Related Threads on Isomorphism between groups

Replies
3
Views
4K
Replies
1
Views
2K
Replies
4
Views
2K
  • Last Post
Replies
4
Views
5K
  • Last Post
Replies
16
Views
9K
Replies
1
Views
2K
  • Last Post
Replies
2
Views
4K
  • Last Post
Replies
4
Views
2K
Replies
15
Views
8K
Replies
7
Views
2K
Top