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i think two more abelian groups are possible .

1 and 3 are given ,2 and 4 also exist .if i m wrong let me know ,thank you.

in Set Theory & Algebra by Boss (10.8k points) | 175 views
Both the groups you found out are isomorphic to the second group given in Rosen.

@Mk Utkarsh

Please explain what is isomorphic group..i have seen this for many times but haven't studied about it..


MiNiPanda added the answer below


Is group theory given in Rosen  Prateek Raghuvanshi ?  I never found it . :(. Can you plz tell me the edition exactly or the reference for reading group theory. Thanks


Pawan Kumar 2 It's in Indian edition 


Thanks brother  Mk Utkarsh

@Prateek,   Can you please explain the mapping for elements b and c from set A to set B in the given answer ?

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Isomorphism : A group homomorphism that is bijective. 

We proved that all 3 groups are homomorphic as well as isomorphic.


Note:  I rearranged the elements of table to show the similarities between all 3 tables.


by Boss (36.5k points)
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@Utkarsh , can you please explain how you created the function mapping between 2 sets. here $f$ is not defined . so how you mapped the element from one set to another. Please explain :)
yes $f$ is not defined, i mapped the elements with same behavior in A to same behavior in B and C to show one to one correspondence in the given groups
Sorry.. I am still not getting the mapping.  Can you please explain a little bit about mapping for elements b And c from set A to set B ?
check the mapping and take group B's table and map the elements based of the mapping $A \rightarrow B$ you'll get the same mapping back of A

or vice versa, take A's table and interchange elements d and c you'll get the same table back.

Behavior of group B's d is same as group A's c


Behavior of group B's c is same as group A's d
got it. Thank you :)
very well explained :D
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