Exercise · ZW-ABELIAN-GROUP-006 · ★★☆ Intermediate
In an abelian group, what is (a∗b)∗a⁻¹ equal to?
Choose one option.
Show answer
- D. b
Explanation
Since G is abelian, a ∗ b = b ∗ a. Then (a ∗ b) ∗ a⁻¹ = (b ∗ a) ∗ a⁻¹ = b ∗ (a ∗ a⁻¹) = b ∗ e = b. Without commutativity the result would be a ∗ b ∗ a⁻¹, which in general is not b.
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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