Exercise · ZW-FIRST-ISOMORPHISM-THEOREM-003 · ★☆☆ Basic
A surjective homomorphism \(\varphi:G\to H\) has \(|G|=24\) and \(|\ker\varphi|=4\). What is \(|H|\)?
Choose one option.
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- D. 6
Explanation
Since \(\varphi\) is surjective, \(H=\operatorname{im}\varphi\cong G/\ker\varphi\), which has \(24/4=6\) elements. Multiplying (96) or subtracting (20) does not give the size of a quotient.
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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