Show that there is no homomorphism from Z 64 onto Z 4 Z 8Sol

Show that there is no homomorphism from Z 64 onto Z 4 Z 8

Solution

Any Quotient Group of a cyclic group is cyclic. Therefore the image of Z64 under a homomorphism is a cyclic group and so cannot be Z4 Z8
And For a homomorphism : Z64 Z4 Z8, (Z64) is a cyclic group generated by (1). But Z4 Z8 is not cyclic, so (Z64) != Z4 Z8. Therefore is not onto.