Let φ:ℤ→ℤ_10 be given by φ(x)=4xbmod 10. Which group is ℤ/kerφ isomorphic to?

Exercise · ZW-FIRST-ISOMORPHISM-THEOREM-006 · ★★☆ Intermediate

Let \(\varphi:\mathbb{Z}\to\mathbb{Z}_{10}\) be given by \(\varphi(x)=4x\bmod 10\). Which group is \(\mathbb{Z}/\ker\varphi\) isomorphic to?

Choose one option.

Show answer

B. \(\mathbb{Z}_5\)

Explanation

Since \(4x\equiv 0 \pmod{10}\) is equivalent to \(x\equiv0\pmod 5\), the kernel is \(5\mathbb{Z}\) and \(\mathbb{Z}/5\mathbb{Z}\cong\mathbb{Z}_5\). It matches the image \(\{0,2,4,6,8\}\), which has 5 elements, not 10.

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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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