《Discrete Mathematics II教学-华南理工》Lecture 8 Suppl Groups of Permutations.pdfVIP

Introduction to Modern Algebra Part II. Permutations, Cosets, and Direct Products II.8. Groups of Permutations September 20, 2015 () Introduction to Modern Algebra September 20, 2015 1 / 9 Table of contents 1 Lemma 2 Theorem 8.5. 3 Lemma 8.15. 4 Theorem 8.16., Cayley’s Theorem () Introduction to Modern Algebra September 20, 2015 2 / 9 Lemma Lemma Lemma. If σ and τ are permutations on set A, then the composite τ σ function σ ◦ τ (defined as A → A → A is a permutation on A. Normally we drop the composition symbol ◦ and write σ ◦ τ = στ . Notice that we must read this from right to left since στ is permutation τ first, followed by permutation σ . Proof. We must only show that στ is one-to-one and onto. () Introduction to Modern Algebra September 20, 2015 3 / 9 Lemma Lemma Lemma. If σ and τ are permutations on set A, then the composite τ σ function σ ◦ τ (defined as A → A → A is a permutation on A. Normally we drop the composition symbol ◦ and write σ ◦ τ = στ . Notice that we must read this from right to left since στ is permutation τ first, followed by permutation σ . Proof. We must only show that στ is one-to-one and onto. For one-to-one (see page 4 for the definition), suppose (στ ) (a ) = (στ ) (a ); that is, 1 2 σ (τ (a )) = σ (τ (a )). Since σ is one-to-one, then the two “inputs” of τ 1 2 must be the same and so a = a . Therefore στ is one to one. 1 2 () Introduction to Modern Algebra September 20, 2015 3 /