I attempt to divide a C8 by C2.
My reasoning: C2 is a normal subgroup of C8. C2 forms 2 cosets: {0,2,4,6} and 1+{0,2,4,6}. C8/C2 isomorphic to C4.
But I know that C2*C4 is an Abelian group. It is not C8.
Theorem 8.7. Cn*Cm=Cnm if and only if n and m are relatively prime.
2 and 4 are not relatively prime. Their multiplication can not form C8.
Where is my mistake?