Is Z3 X Z5 isomorphic to Z15?

  • Thread starter margaret23
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In summary: Both groups are isomorphic if and only if the order of each element in Z3xZ5 is the same as the order of each element in Z15.
  • #1
margaret23
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Show that Z3 X Z5 is isomorphic to Z15 (where Zn is like the intergers mod n)

i m not sure if i m proving it right.. if i first right out Z3 x Z5 ={(1x1), (1X2),(1x3),(1x4), (1X5), (2X1),(2x2),(2x3), (2x4), (2X5), (3x1), (3x3),(3x3),(3x4), (3X5)}
Z15={1,2,3...15}

both of which have 15 elements... there for they are in the same form?? therefore isomorphic??

i would appreciate any help
thanks
 
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  • #2
To show that Z3 X Z5 is isomorphic to Z15 you have to come up with a one-to-one onto function between the two. I think something like this might work if (a,b) is in Z3 X Z5 then the mapping p^a*q^b where p and q are unique primes is one-to-one by the unique factorization thm. However, it's not onto but we do know it's well ordered so we have elements a_1<a_2<...<a_15 then you can take a_n to n and this would make it onto.
 
  • #3
buzzmath said:
To show that Z3 X Z5 is isomorphic to Z15 you have to come up with a one-to-one onto function between the two. I think something like this might work if (a,b) is in Z3 X Z5 then the mapping p^a*q^b where p and q are unique primes is one-to-one by the unique factorization thm. However, it's not onto but we do know it's well ordered so we have elements a_1<a_2<...<a_15 then you can take a_n to n and this would make it onto.

To show they're isomorphic
You also need to show that if F is the mapping between them, * is the operation on Z3xZ5, and # is the operation on Z15

that
F(a*b) = F(a)#F(b)
where a and b are elements of Z3xZ5 if the function maps Z3xZ5 to Z15.
 
  • #4
Sorry, I forgot to say that the mapping also has to be a homomorphism.
 
  • #5
margaret23 said:
Show that Z3 X Z5 is isomorphic to Z15 (where Zn is like the intergers mod n)

i m not sure if i m proving it right.. if i first right out Z3 x Z5 ={(1x1), (1X2),(1x3),(1x4), (1X5), (2X1),(2x2),(2x3), (2x4), (2X5), (3x1), (3x3),(3x3),(3x4), (3X5)}
Z15={1,2,3...15}

both of which have 15 elements... there for they are in the same form?? therefore isomorphic?

The number of elements does not determine the group. This is clear, since abelian groups cannot be isomorphic to non-abelian groups.

It is easier to map elements of Z15 to the other group, as it happens, from the description you've given.

Of course, if you can show that some element of Z3xZ5 has order 15 you're also done, but do you understand why?
 

1. What does it mean for a number to be isomorphic?

Isomorphism is a mathematical concept that refers to the structural similarity between two objects. In the context of numbers, an isomorphic number is one that can be transformed into another number by rearranging its digits. In other words, the numbers have the same underlying structure, even though they may appear different.

2. How can I tell if two numbers are isomorphic?

To determine if two numbers are isomorphic, you can compare the digits in each number and see if they can be rearranged to form the other number. For example, the numbers 123 and 231 are isomorphic because they have the same digits, just in a different order.

3. Are all numbers isomorphic to each other?

No, not all numbers are isomorphic to each other. For two numbers to be isomorphic, they must have the same number of digits and the same digits, just in a different order. This means that numbers with different numbers of digits or different digits are not isomorphic.

4. Can fractions or decimals be isomorphic?

Yes, fractions and decimals can also be isomorphic. For example, the fractions 1/2 and 3/6 are isomorphic because they can be rearranged to form each other. Similarly, the decimals 1.5 and 5.1 are also isomorphic.

5. What is the significance of isomorphic numbers?

Isomorphic numbers may not have any practical significance, but they are interesting from a mathematical standpoint. They demonstrate the concept of structural similarity and can be found in various mathematical patterns and sequences. Additionally, isomorphic numbers are used in certain encryption algorithms to create secure codes.

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