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charlamov
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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
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
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.
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.
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.
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.
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.