Exercise · ZW-FIRST-ISOMORPHISM-THEOREM-004 · ★☆☆ Basic
If \(\varphi:G\to H\) is a homomorphism with \(\ker\varphi=\{e\}\), the first isomorphism theorem says that \(G\) is isomorphic to…
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- B. the image of \(\varphi\)
Explanation
With trivial kernel, \(G/\{e\}\cong G\), so \(G\cong\operatorname{im}\varphi\). It would be all of \(H\) only if \(\varphi\) were also surjective.
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-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-008 Let φ:ℤ→ℤ_12 be given by φ(x)=8xbmod12. What is the image of φ?
- ZW-FIRST-ISOMORPHISM-THEOREM-009 Let φ:ℤ_36→ℤ_36 be given by φ(x)=9x. What is the pair (|kerφ|, |imφ|)?
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