Exercise · ZW-ABELIAN-GROUP-004 · ★★☆ Intermediate
To prove that G is abelian one needs to verify...
Choose one option.
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- B. The equality for all a,b ∈ G
Explanation
To prove that G is abelian we must verify a ∗ b = b ∗ a for every pair a, b ∈ G. Checking one case is not enough, and G need not be finite.
Review the concept: Abelian group
A group is abelian, or commutative, if its operation allows the order of any two elements to be swapped.
\[a\ast b=b\ast a \quad \forall a,b\in G\]
More practice problems: abelian group
- ZW-ABELIAN-GROUP-001 An abelian group is a group whose operation is...
- ZW-ABELIAN-GROUP-002 Which is an abelian group?
- ZW-ABELIAN-GROUP-003 Commutativity means that...
- ZW-ABELIAN-GROUP-005 Every abelian group is necessarily...
- ZW-ABELIAN-GROUP-006 In an abelian group, what is (a∗b)∗a⁻¹ equal to?
- ZW-ABELIAN-GROUP-007 If in a group G there are elements a and b with a∗b ≠ b∗a, what can be concluded?
- ZW-ABELIAN-GROUP-008 S₃ is not abelian because...
- ZW-ABELIAN-GROUP-009 Select the only abelian group with its usual operation.