Exercise · ZW-FIRST-ISOMORPHISM-THEOREM-008 · ★★☆ Intermediate
Let \(\varphi:\mathbb{Z}\to\mathbb{Z}_{12}\) be given by \(\varphi(x)=8x\bmod12\). What is the image of \(\varphi\)?
Choose one option.
Show answer
- D. \(\{0,4,8\}\)
Explanation
We have \(8x\equiv0\pmod{12}\) if and only if \(3\mid x\), so \(\ker\varphi=3\mathbb{Z}\) and the image is isomorphic to \(\mathbb{Z}/3\mathbb{Z}\cong\mathbb{Z}_3\), with 3 elements: \(0,8,16\equiv4\).
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
- ZW-FIRST-ISOMORPHISM-THEOREM-001 Let φ:ℤ_12→ℤ_12 be given by φ(x)=3x. How many elements does the image of φ have?
- ZW-FIRST-ISOMORPHISM-THEOREM-002 The determinant det:GL_2(ℝ)→ℝ^* is a surjective group homomorphism with kernel SL_2(ℝ). What does t…
- ZW-FIRST-ISOMORPHISM-THEOREM-003 A surjective homomorphism φ:G→ H has |G|=24 and |kerφ|=4. What is |H|?
- ZW-FIRST-ISOMORPHISM-THEOREM-004 If φ:G→ H is a homomorphism with kerφ=\e\, the first isomorphism theorem says that G is isomorphic…
- ZW-FIRST-ISOMORPHISM-THEOREM-005 The function exp:(ℝ,+)→(ℝ^*,·), xmapsto e^x, is an injective homomorphism with image ℝ_>0. Which gr…
- ZW-FIRST-ISOMORPHISM-THEOREM-006 Let φ:ℤ→ℤ_10 be given by φ(x)=4xbmod 10. Which group is ℤ/kerφ isomorphic to?
- ZW-FIRST-ISOMORPHISM-THEOREM-007 Let φ:ℤ_15→ℤ_15 be given by φ(x)=6x. How many elements does kerφ have?
- ZW-FIRST-ISOMORPHISM-THEOREM-009 Let φ:ℤ_36→ℤ_36 be given by φ(x)=9x. What is the pair (|kerφ|, |imφ|)?
← ZW-FIRST-ISOMORPHISM-THEOREM-007 ZW-FIRST-ISOMORPHISM-THEOREM-009 →