Symmetric and Alternating Groups

Abstract Algebra · Axiom Academy

LESSON Symmetric and Alternating Groups Understanding permutation groups through cycle notation, order calculations, and the distinction between even and odd permutations. A permutation can be thought of as a rearrangement. For example, in S 4 , we might send 1→3, 2→1, 3→4, 4→2. We can write this in two-line notation: Instead of two-line notation, we can write permutations more compactly using cycle notation . A cycle (a b c) means: a→b, b→c, c→a. The order of a permutation σ is the smallest positive integer k such that σ k = e (the identity). This depends on the cycle structure! A transposition is a cycle of length 2, swapping two elements. Every permutation can be written as a product of transpositions. The collection of all even permutations forms a subgroup of S n called the alternating group A n . Contains exactly the even permutations in S n Size: |A n | = n!/2 (exactly half of S n ) Important: A 5 is the smallest non-abelian simple group!

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