Question

What is the condition for a group \( G \) to be Abelian based on the commutator subgroup?

• A. Commutator subgroup is equal to \(G\)
• B. Commutator subgroup is trivial
• C. Commutator subgroup is infinite
• D. Commutator subgroup is cyclic

Hint:

Group is Abelian \( \Leftrightarrow \) Commutator subgroup is trivial

Answer 1

The Correct Option is B

Concept: Commutator Subgroup
The commutator of two elements \(a, b \in G\) is: \[ [a,b] = aba^{-1}b^{-1} \] The commutator subgroup \(G'\) (or \([G,G]\)) is generated by all such commutators.
Step 1: Abelian group definition
A group is Abelian if: \[ ab = ba \quad \forall a,b \in G \]
Step 2: Effect on commutators
If \(ab = ba\), then: \[ [a,b] = e \] (identity element)
Step 3: Conclusion
All commutators are identity \( \Rightarrow \) commutator subgroup is trivial: \[ G' = \{e\} \]