search
Log In
0 votes
303 views

reference:Rosen

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 303 views
1
Both the groups you found out are isomorphic to the second group given in Rosen.
0

@Mk Utkarsh

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

0

MiNiPanda added the answer below

0

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

0

Pawan Kumar 2 It's in Indian edition 

0

Thanks brother  Mk Utkarsh

0
@Prateek,   Can you please explain the mapping for elements b and c from set A to set B in the given answer ?

1 Answer

3 votes
 
Best answer

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.

 


selected by
0
@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 :)
0
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
0
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 ?
1
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

and

Behavior of group B's c is same as group A's d
0
got it. Thank you :)
0
very well explained :D

Related questions

0 votes
0 answers
1
127 views
Is Every Group of Order $P^{k}$ such that P is prime and K is positive integer ABELIAN
asked Dec 26, 2018 in Set Theory & Algebra jatin khachane 1 127 views
0 votes
1 answer
2
393 views
Consider the following statements: $\mathbf{S_1:}\;\;$If a group $\mathbf{(G,*)}$ is of order $\mathbf n$ and $\mathrm {a \in G}$ is such that $\mathrm {a^m=e}$ for some integer $\mathrm {m \le n}$ then $\mathbf m$ must divide $\mathbf n$. $\mathbf {S_2:}\;\;$If a ... $(3)\;\;\;\text{Niether}\; \mathrm{S_1}\;\text{nor}\;\mathrm{S_2}$
asked Dec 29, 2019 in Set Theory & Algebra Sanjay Sharma 393 views
0 votes
1 answer
3
309 views
if (G,*) is a cyclic group of order 97 , then number of generator of G is equal to ___
asked Jan 16, 2019 in Set Theory & Algebra Rahul_Rathod_ 309 views
0 votes
0 answers
4
237 views
Is this monoid: Addition modulo (take mode using m) on the set of Integers (Z m)={0,1,2,3,4,…..m-1} i.e. For all a a (+ modulo using m) e = e (+ modulo using m) a =a here, e is an identity element
asked Jan 14, 2019 in Set Theory & Algebra Nandkishor3939 237 views
...