- #1
Dahaka14
- 73
- 0
Homework Statement
Show that [tex]\mathbb{Z}_{8}[/tex] is not isomorphic to [tex]\mathbb{Z}_{4}\times\mathbb{Z}_{2}[/tex].
Homework Equations
[tex]\mathbb{Z}_{mn}\cong\mathbb{Z}_{m}\times\mathbb{Z}_{n}\iff \gcd(m,n)=1[/tex]
The Attempt at a Solution
I would say that since [tex] \gcd(4,2)\neq1[/tex], they are not isomorphic.
There must be something that I am misunderstanding though since [tex]\mathbb{Z}_{4}[/tex] is not isomorphic [tex]\mathbb{Z}_{2}\times\mathbb{Z}_{2}[/tex]...