Exercise · ZW-KERNEL-OF-A-HOMOMORPHISM-008 · ★★★ Advanced
How many homomorphisms \(\varphi\colon\mathbb{Z}_{12}\to\mathbb{Z}_{18}\) have a kernel with exactly \(2\) elements?
Choose one option.
Show answer
- C. 2
Explanation
Each is determined by \(a=\varphi(1)\) with \(12a\equiv0\pmod{18}\), i.e. \(a\in\{0,3,\dots,15\}\). The kernel has \(2\) elements when \(a\) has order \(6\): \(a=3\) and \(a=15\).
Review the concept: Kernel of a homomorphism
The kernel of a group homomorphism \(\varphi\colon G\to H\) is the set of elements of \(G\) that \(\varphi\) sends to the identity \(e_H\) of \(H\).
\[\ker\varphi=\{g\in G:\ \varphi(g)=e_H\}\]
More practice problems: kernel of a homomorphism
- ZW-KERNEL-OF-A-HOMOMORPHISM-001 Let φcolonℤ_12→ℤ_4 be given by φ(x)=x bmod 4. What is the kernel of φ?
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- ZW-KERNEL-OF-A-HOMOMORPHISM-003 Which element belongs to the kernel of every homomorphism φcolon G→ H?
- ZW-KERNEL-OF-A-HOMOMORPHISM-004 Let φcolonℤ_18→ℤ_6 with φ(x)=4x bmod 6. How many elements does its kernel have?
- ZW-KERNEL-OF-A-HOMOMORPHISM-005 Let φcolon(ℤ_7^*,·)→(ℤ_7^*,·) with φ(x)=x^2. What is its kernel?
- ZW-KERNEL-OF-A-HOMOMORPHISM-006 A homomorphism φcolon G→ H is injective. What can be stated for sure?
- ZW-KERNEL-OF-A-HOMOMORPHISM-007 Let φcolonℤ_8→ℤ_8 with φ(x)=6x bmod 8. How many distinct elements does its image have?
- ZW-KERNEL-OF-A-HOMOMORPHISM-009 A group G of order 15 maps onto ℤ_5 by a surjective homomorphism. How many elements does the kernel…
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