Let φ:ℤ_36→ℤ_36 be given by φ(x)=9x. What is the pair (|kerφ|, |imφ|)?

Exercise · ZW-FIRST-ISOMORPHISM-THEOREM-009 · ★★★ Advanced

Let \(\varphi:\mathbb{Z}_{36}\to\mathbb{Z}_{36}\) be given by \(\varphi(x)=9x\). What is the pair \((|\ker\varphi|,\,|\operatorname{im}\varphi|)\)?

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C. \((9,4)\)

Explanation

We have \(9x\equiv0\pmod{36}\) if and only if \(4\mid x\), giving 9 elements in the kernel; the image \(\{0,9,18,27\}\) has 4. The sizes satisfy \(36=|\ker\varphi|\cdot|\operatorname{im}\varphi|\), but the order of the pair matters.

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Review the concept: First isomorphism theorem

Every group homomorphism \(\varphi:G\to H\) induces an isomorphism between the quotient of \(G\) by the kernel of \(\varphi\) and the image of \(\varphi\).

\[G/\ker\varphi\;\cong\;\varphi(G),\qquad a\ker\varphi\mapsto\varphi(a)\]

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