The document discusses permutations, which are rearrangements of objects in a set. A permutation is defined as the number of ways objects in a set can be uniquely ordered. The number of permutations of a set of n objects is n!. Permutations form groups under composition, with the set of all permutations of a set forming a symmetric group. Permutations have practical applications in areas like computer networking and cryptography.

1. Permutations

2. Permutations Suppose a set consists of a finite number of distinct objects. A permutation of these objects is a rearrangement of them among themselves. The permutation of a set is the number of ways that the items in the set can be uniquely ordered.

3. Permutations The set of red, white, blue and green counters could be permuted by replacing the green counter by a blue one and vice versa. The permutations of the set { 1, 2, 3} are { 1, 2, 3}, { 1, 3, 2}, { 2, 1, 3}, { 2, 3, 1}, { 3, 1, 2} and { 3, 2, 1}.

4. Permutations In practice it is easier if we label the objects 1,2,3,4. The permutation that replaces the green counter with blue would then be replace 1,2,3,4 by 1,2,4,3. We write this permutation . where each number is replaced by the number underneath.

5. Permutations Aside from theoretical interest in set theory, permutations have some practical use. Permutations can be used to define switching networks in computer networking and parallel processing (see Figure 1). Permutation also used in a variety of cryptographic algorithms. y Figure 1, An Omega permuation network, from Interconnection Networks: an Engineering approach . by Duato, Yalamanchili and Ni, Morgan Kaufmann, 2003, Pg. 32

6. Permutations For n objects, the number of permutations is n ! The product AB of two permutations A and B is defined as B followed by A .

7. Permutations If A is the permutation of the set {1,2,3,4} and it maps 1->3, 2 ->4, 3 ->2, 4 ->1, then it can be represented by . If B is the permutation of the set {1,2,3,4} and it maps 1 ->2,2 ->1, 3 ->4,4 ->3, then it can be represented by .

8. Permutations Then the product AB = Which can also be represented by . Note: we start with the right set and then map to the left.

9. Permutations Then the product AB = Which can also be represented by . Note: we start with the right set and then map to the left.

10. Permutations Then the product AB = That is 1->4,2 ->3,3 ->1,4 ->2. Note: some websites/books map bottom to top, rather than top to bottom.

11. Permutations Writing C = AB we can produce a Cayley table for the set of permutations X = { I,A,B,C } where I = .

12. Permutations Complete the operation table for { I,A,B,C }. I A B C I A B C

13. Permutations Complete the operation table for { I , A,B,C }. I A B C I I A B C A A B C I B B C I A C C I A B

14. Permutations The set X is closed under composition of permutations. Composition of mappings of any type is associative, this operation is associative. X contains the identity I , and each element has an inverse in X . Hence X forms a group. Note: not every pairing of permutations forms a group, only some, so it is essential that you check if it forms a group.

15. Permutations , is the set of all permutations of the set {1,2,3,4} and forms a group. The associative law for composition of permutations holds. Since contains all possible permutations, it contains the identity and the product of any given pair, so it is closed. Given that the inverse is another permutation, then for all permutations the inverse is also in .

16. Inverse Permutations If X is the permutation Then Note: composition of permutations is not usually commutative. forms a non-Abelian group.

17. Symmetric Groups The order of the permutation group is 4! or 24. In general, the set of all permutations of a set with n elements is the symmetric group of order n !

18. Exercises Exercise 19.3