Isomorphism between groups

In summary, it is true that if there is a monomorphism from group A to group B and a monomorphism from group B to group A, then A and B are isomorphic. This is true for finite groups, but not necessarily true for infinite groups. A good strategy for determining the truth of this statement is to try constructing a counterexample. One example is the free (non-abelian) groups on different sets of generators, which can be seen through the construction of a covering space. This was first noted by Eilenberg in a lecture 30-40 years ago and can be found in Hatcher's free algebraic topology book. Additionally, the free group on two generators contains a subgroup that is isomorphic
  • #1
charlamov
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
 
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  • #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.
 
  • #3
thanks, i finally have found that it is not true generally
 
  • #4
did you find the counterexample?
 
  • #5
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
 
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  • #6
mathwonk said:
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
 

1. What is an isomorphism between groups?

An isomorphism between groups is a bijective map between two groups that preserves the group structure. In other words, it is a function that maps elements from one group to another in a way that respects the group operations and the identity elements.

2. How do you prove two groups are isomorphic?

To prove two groups are isomorphic, you need to show that there exists a bijective map between the two groups that preserves the group operations. This can be done by explicitly constructing the map and showing that it satisfies the properties of an isomorphism, or by using other techniques such as showing that the groups have the same group table.

3. Can two non-isomorphic groups have the same number of elements?

Yes, it is possible for two non-isomorphic groups to have the same number of elements. For example, the groups of even and odd integers under addition have the same number of elements, but they are not isomorphic as their group structures are different.

4. What is the significance of isomorphisms between groups?

Isomorphisms between groups allow us to study and understand different groups by relating them to each other. By showing that two groups are isomorphic, we can use knowledge and techniques from one group to solve problems in the other group.

5. Are all isomorphic groups equal?

No, isomorphic groups are not necessarily equal. While they have the same structure and behave in a similar way, they may have different elements and operations. It is important to note that isomorphism is a relation between groups, not an equality statement.

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