Exercise · ZW-ABELIAN-GROUP-010 · ★★★ Advanced
Let G be a group in which (a∗b)∗(a∗b) = (a∗a)∗(b∗b) for all a, b ∈ G. What can be concluded?
Choose one option.
Show answer
- D. G is abelian
Explanation
Writing a∗b∗a∗b = a∗a∗b∗b and cancelling a on the left and b on the right leaves b ∗ a = a ∗ b for all a, b: G is abelian. It need not be finite or cyclic: (ℝ, +) satisfies the hypothesis and is neither.
Review the concept: Abelian group
A group is abelian, or commutative, if its operation allows the order of any two elements to be swapped.
\[a\ast b=b\ast a \quad \forall a,b\in G\]
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