Given α:Z9×Z27→Z27 be given by α(a,b)=3b for aϵZ9,bϵZ27 and
(a,b)+(c,d)=(a+c,b+d) .
(i) Homomorphism: α((a,b)+(c,d))=α(a+c,b+d)=2(b+d)=2b+2d
Also, α(a,b)+α(c,d)=2b+2d .
Hence, α((a,b)+(c,d))=α(a,b)+α(c,d) .
Hence, α is a Homomorphism map.
(ii) ker(α)={(a,b):α(a,b)=0}
α(a,b)=0⟹2b=0⟹b=27n where n∈Z .
So, ker(α)={(a,27n):n∈Z}
Comments