Exercise · ZW-ABELIAN-GROUP-008 · ★★☆ Intermediate
S₃ is not abelian because...
Choose one option.
Show answer
- B. There are permutations that do not commute
Explanation
S₃ is not abelian because there are permutations that do not commute: for example, composing a transposition with a 3-cycle gives different results depending on the order. It is still a group, with identity and inverses.
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-004 To prove that G is abelian one needs to verify...
- 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-009 Select the only abelian group with its usual operation.