Exercise · ZW-FIRST-ISOMORPHISM-THEOREM-001 · ★☆☆ Basic
Let \(\varphi:\mathbb{Z}_{12}\to\mathbb{Z}_{12}\) be given by \(\varphi(x)=3x\). How many elements does the image of \(\varphi\) have?
Choose one option.
Show answer
- C. 4
Explanation
The kernel is \(\{0,4,8\}\), with 3 elements. By the first theorem, \(|\operatorname{im}\varphi|=12/3=4\) (the image is \(\{0,3,6,9\}\)). The 3 is the size of the kernel, not of the image.
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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