Comparing Elements of Order 4 in External Direct Products

[mehtamonica]

Homework Statement

Explain why external direct products z8 + z4 and z80000000+ z4000000 have same number of elements of order 4?

Homework Equations

The Attempt at a Solution

Z 8 = { 0, 1, 2, 3, 5, 6, 7, } , order of elements : 0 =1, 1=8, 2=4 , 3=8, 4=2, 5=8, 6=4, 7=8.

Z 4= { 0, 1, 2, 3} order of elements : 0= 1, 1=4, 2=2, 3=4.

Hence, in the EDP Z8 + Z4...it seems that there are 12 elements of order 4

(0, 1), (0, 3), (2,0) (2, 1), (2,2) (2, 3) (4,1) (4,3) (6,0) (6,1) (6,2) (6, 3).

Please explain how to count the no. of elements of order 4 in z80000000+ z4000000

 

[micromass]

Can you use the fundamental theorem of abelian groups to write $\mathbb{Z}_{80000000}$ and $\mathbb{Z}_{4000000}$ differently??