Question

plz find homomorphism z8 to Q8

Answer 1

Any homomorphism into an abelian group factors through the abelianisation which is Qab8≅C2×C2Q8ab≅C2×C2 so in particular its image lies in the subgroup of elements of order dividing 2. So you are really asking about Hom(C2×C2,C4)Hom⁡(C2×C2,C4) and Hom(C2×C2,C8)Hom⁡(C2×C2,C8) but in both cases homomorphisms land in a cyclic subgroup of order 2 so you want Hom(C2×C2,{1,−1})≅C2×C2Hom⁡(C2×C2,{1,−1})≅C2×C2 where I am using the isomorphism C2≅{1,−1}C2≅{1,−1} .

The three non-trivial homomorphisms do the following on i,j,k:

i,j -> -1; i,k -> -1; j,k -> -1