Exercise · ZW-KERNEL-OF-A-HOMOMORPHISM-005 · ★★☆ Intermediate
Let \(\varphi\colon(\mathbb{Z}_7^{*},\cdot)\to(\mathbb{Z}_7^{*},\cdot)\) with \(\varphi(x)=x^2\). What is its kernel?
Choose one option.
Show answer
- B. \(\{1,6\}\)
Explanation
The identity is \(1\). We need \(x^2\equiv1\pmod 7\): only \(x=1\) and \(x=6\) work (\(6^2=36\equiv1\)).
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?
- ZW-KERNEL-OF-A-HOMOMORPHISM-002 Let φcolon(ℝ,+)→(ℝ_>0,·) with φ(x)=e^x. What is its kernel?
- 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-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-008 How many homomorphisms φcolonℤ_12→ℤ_18 have a kernel with exactly 2 elements?
- 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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