Question

Can there be a homomorphism from Z₈ ⊕Z₂ onto Z₄⊕Z₄ ? Give reasons for your

answer.

Answer 1

Let us prove that there is no homomorphism from $\Z_8 \otimes\Z_2$ onto $\Z_4\times\Z_4$ using method of contradiction. Assume that $\varphi:\Z_8 \otimes\Z_2 \to \Z_4 \otimes\Z_4$ is a surjective homomorphism. Since $|\Z_8 \otimes\Z_2|=16=|\Z_4 \otimes\Z_4 |,$ we conclude that the surjection $\varphi$ is a bijection, and hence $\varphi:\Z_8 \otimes\Z_2 \to \Z_4 \otimes\Z_4$ is an isomomorhism. Taking into account that isomorphisms preserve orders of elements and $\Z_8 \otimes\Z_2$ contains the element $(1,0)$ of order 8, but all elements of $\Z_4 \otimes\Z_4$ have order less or equal to 4, we conclude that our assumption is not true. Consequently, there is no homomorphism from $\Z_8 \otimes\Z_2$ onto $\Z_4\times\Z_4.$