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

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Z64 is a cyclic group generated by 1, so its image under any homomorphism must also be cyclic. However, Z4 x Z8 is not cyclic, so there cannot exist a homomorphism from Z64 onto Z4 x Z8. Any homomorphism from Z64 cannot have an image of Z4 x Z8 since the image must be cyclic and Z4 x Z8 is not cyclic.

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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.

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