Question

Determine all homomorphisms from z12 to z30

Answer 1

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Answer 2

Given:

z12 to z30.

To find:

homomorphisms from z12 to z30.

Solution:

1)Let us take Ф: Z12 → Z30  be an homomorphism from z12 to z30.

2)We know that  Ф  is fully determined by the value of  Ф(1) ,

because ,  Ф(n)=n⋅Ф(1)     ∀n∈Z12.  

3)If we take n=30 ,

• we get that,

• Ф(30)=30⋅Ф(1)=0.  

• But, on the other hand, we have that

• Ф(30)=Ф(6)=6⋅Ф(1).  

• From  6⋅Ф(1)=0  

• we conclude that  Ф(1)  must be a multiple of  5 .

4)So the functions are of the form  x↦5k⋅x , where  k∈{0,1,2,3,4,5} .

• Any such function is well defined, because

• Ф(n+12t)=5k⋅(n+12t)=5kn+60kt=5kn .

5)It will be very easy to show that these functions preserve the addition operation. So, we need to decide which of them also preserve multiplication. Note that

• 5k=Ф(1)=Ф(1⋅1)=Ф(1)Ф(1)=25k2.  

• So  25k2−5k  is a multiple of  30 and  5k2−k  is a multiple of  6 .

• The values of  k  for which this is true are  k∈{0,2,3,5} .

• For any of these values, we have that

• Ф(a)Ф(b)−Ф(ab)=25k2ab−5kab=5ab(5k2−k)  is a multiple of  30.

So we have  4  homomorphisms