Question #123763
Let α : Z₉* Z₂₇→ Z₂₇ be given by α ((a, b)) = 3b for a ϵ Z₉, b ϵ Z₂₇ and
(a, b) + (c, d) = (a + c, b + d) is given by addition modulo n for each group Zn.
i. Show that α is a homomorphism.
ii. Find Ker(α).
1
Expert's answer
2020-06-29T18:16:30-0400

Given α:Z9×Z27Z27\alpha : Z₉ \times Z₂₇→ Z₂₇ be given by α(a,b)=3b\alpha (a, b) = 3b for aϵZ9,bϵZ27a ϵ Z_9, b ϵ Z_{27} and

(a,b)+(c,d)=(a+c,b+d)(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\alpha((a, b) + (c, d)) = \alpha(a + c, b + d) = 2(b+d) = 2b + 2d

Also, α(a,b)+α(c,d)=2b+2d\alpha(a,b)+\alpha(c,d) = 2b+2d .

Hence, α((a,b)+(c,d))=α(a,b)+α(c,d)\alpha((a, b) + (c, d)) = \alpha(a, b) + \alpha(c, d) .

Hence, α\alpha is a Homomorphism map.

(ii) ker(α)={(a,b):α(a,b)=0}ker(\alpha) = \{(a,b): \alpha(a,b)=0 \}

α(a,b)=0    2b=0    b=27n\alpha(a,b)=0 \implies 2b = 0 \implies b = 27 n where nZn\in \Z .

So, ker(α)={(a,27n):nZ}ker(\alpha) = \{(a,27n): n \in \Z\}


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