No, the S4 (the symmetric group on 4 elements) is not abelian.
The S4 is a permutation group consisting of all possible permutations of a set of four distinct elements. To understand why it's not abelian, it's essential to first define what an abelian group is.
Understanding Abelian Groups
An abelian group is a fundamental concept in abstract algebra. Named after the Norwegian mathematician Niels Henrik Abel, a group is considered abelian if its group operation is commutative.
Key characteristics of an abelian group include:
- Commutativity: For any two elements, 'a' and 'b', within the group, the result of their operation 'a b' must be equal to 'b a'. This means the order in which the elements are combined does not affect the outcome.
- Associativity: While also a property of all groups, it means (a b) c = a (b c).
- Identity Element: There exists an identity element 'e' such that for every element 'a' in the group, 'e a = a e = a'.
- Inverse Element: For every element 'a' in the group, there exists an inverse element 'a⁻¹' such that 'a a⁻¹ = a⁻¹ a = e'.
Why S4 Is Not Abelian
The symmetric group S4 contains 24 distinct permutations (4! = 24). For a group to be abelian, all pairs of elements must commute. However, permutation groups like S4 quickly demonstrate non-commutative behavior as the number of elements grows.
For instance, when considering specific permutations within S4, the order of operations significantly matters. If we take two particular permutations, let's denote them as 'r1' and 'd3', their composition 'r1 d3' results in a certain permutation, which can be identified as 'c0'. Conversely, if we reverse the order of composition to 'd3 r1', the result is a different permutation, 'd0'. Since 'c0' is not equal to 'd0', this single example is sufficient to prove that the operations within S4 are not commutative. Therefore, S4 fails the core requirement for being an abelian group.
This non-commutative property is common in larger symmetric groups (S_n for n ≥ 3), making them fundamental examples of non-abelian groups in the study of abstract algebra.
