Exercise · ZW-FIRST-ISOMORPHISM-THEOREM-007 · ★★☆ Intermediate
Let \(\varphi:\mathbb{Z}_{15}\to\mathbb{Z}_{15}\) be given by \(\varphi(x)=6x\). How many elements does \(\ker\varphi\) have?
Choose one option.
Show answer
- A. 3
Explanation
The condition \(6x\equiv0\pmod{15}\) is equivalent to \(2x\equiv0\pmod5\), that is, \(x\in\{0,5,10\}\). The image is \(\{0,3,6,9,12\}\), with 5 elements, and \(15/5=3\) confirms the size of the kernel.
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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