Answer to Question #123763 in Abstract Algebra for Nii Laryea

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 "\\alpha : Z\u2089 \\times Z\u2082\u2087\u2192 Z\u2082\u2087" be given by "\\alpha (a, b) = 3b" for "a \u03f5 Z_9, b \u03f5 Z_{27}" and

"(a, b) + (c, d) = (a + c, b + d)" .

(i) Homomorphism: "\\alpha((a, b) + (c, d)) = \\alpha(a + c, b + d) = 2(b+d) = 2b + 2d"

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

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

Hence, "\\alpha" is a Homomorphism map.

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

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

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


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