6.3 Permutation groups and cyclic groups Exampl ole: Consider the equilateral triangle with vertices 1, 2, and3. Letl1, l2, and l3 be the angle bisectors of the corresponding angles, and leto be their point of intersection a Counterclockwise rotation of the triangle about through120°,240°,360°(0°)
6.3 Permutation groups and cyclic groups Example: Consider the equilateral triangle with vertices 1，2，and 3. Let l1 , l2 , and l3 be the angle bisectors of the corresponding angles, and let O be their point of intersection。 Counterclockwise rotation of the triangle about O through 120°,240°,360° (0°)
f2:1→>2,2→)3,3-)1 nf:1-)3,2-1,3->2 f1:1-1,2->2,3-)3 reflect the linesli,l2, and l3. g1:1-1,2->3,3->2 g2:13,2->2,3-)1 ng3:l->2,2-1,3->3
f2 :1→2，2→3，3→1 f3 :1→3，2→1，3→2 f1 :1→1，2→2，3→3 reflect the lines l1 , l2 , and l3 . g1 :1→1，2→3，3→2 g2 :1→3，2→2，3→1 g3 :1→2，2→1，3→3
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 fand g be two permutations of s. Then the composition of f and g is a permutation of S (2) Letf be a permutation of s. Then the inverse of f 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: Lets 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
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
Theorem 6.10: let s be a finite set with n elements. Then A(S) has n! elements. Definition 10: The group S, is the set of permutations of the first n natural numbers. The group is called the symmetric group on n letters, is called also the permutation group
Theorem 6.10: Let S be a finite set with n elements. Then A(S) has n! elements. Definition 10: The group Sn is the set of permutations of the first n natural numbers. The group is called the symmetric group on n letters, is called also the permutation group