6.3.1 Permutation groups Definition 9: A bijection from a set S to itself is called a permutation of s Lemma 6.1: lets be a set (1) Let and g be two permutations of s. Then the composition off and g is a permutation of s. (2) Let f be a permutation of s. Then the inverse off is a permutation of s
6.3.1 Permutation groups Definition 9: A bijection from a set S to itself is called a permutation of S Lemma 6.1：Let S be a set. (1) Let f and g be two permutations of S. Then the composition of f and g is a permutation of S. (2) Let f be a permutation of S. Then the inverse of f is a permutation of S
Theorem 6.9: Let s be a set. The set of all permutations of S, under the operation of composition of permutations, forms a group A(S) Proof: Lemma 6.1 implies that the rule of multiplication is well-defined associative the identity function from S to s is identity element The inverse permutation g off is a permutation of s
Theorem 6.9：Let S be a set. The set of all permutations of S, under the operation of composition of permutations, forms a group A(S). Proof: Lemma 6.1 implies that the rule of multiplication is well-defined. associative. the identity function from S to S is identity element The inverse permutation g of f is a permutation of S