MATH 113 Lecture Notes - Lecture 7: Bijection, Permutation Group, Computer-Aided Technologies
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Then sx stands for the binary structure of all bijections of x onto x under the composition #. Let a be a set, let sa be the collection of all permutations of a. Then sa is a group under permutation multiplication. Suppose x = {1, 2, 3, 4, , n}. Every element g of sx can be uniquely represented by. Typically, if x is finite, sx is called th permutation group sn. For f and g, it"s just f(g(1)) instead of g(1). Just a double permutation, look in the book for this one. Recall that if a group has exactly 2 elements, then it is isomorphic to z2. Thus s2 is isomorphic to z2 and is abelian. Note that s3 is isomorphic to d3, the symmetries of an equilateral triangle. Every group g is isomorphic to a group of permutations some subgroup of sx for some. Def: suppose % is a permutation of sn.
