Exercise · ZW-FIRST-ISOMORPHISM-THEOREM-010 · ★★★ Advanced
Let \(\varphi:G\to H\) be a group homomorphism. Which of the following statements is false in general?
Choose one option.
Show answer
- B. \(G/\ker\varphi\cong H\)
Explanation
The quotient is isomorphic to the image, which is a subgroup of \(H\) and equals \(H\) only when \(\varphi\) is surjective. For example, the inclusion \(\mathbb{Z}\to\mathbb{Q}\) has \(\mathbb{Z}/\{0\}\cong\mathbb{Z}\not\cong\mathbb{Q}\).
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 \(\varphi\).
\[G/\ker\varphi\;\cong\;\varphi(G),\qquad a\ker\varphi\mapsto\varphi(a)\]
More practice problems: first isomorphism theorem
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- ZW-FIRST-ISOMORPHISM-THEOREM-006 Let φ:ℤ→ℤ_10 be given by φ(x)=4xbmod 10. Which group is ℤ/kerφ isomorphic to?
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