By F. Goodman

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**Sample text**

Then p divides one of the factors. ■ Proof. 12. 18. The prime factorization of a natural number is unique. Proof. 6. DIVISIBILITY IN THE INTEGERS ✐ 33 where the qi ’s and pj ’s are prime and q1 Ä q2 Ä Ä qr and p1 Ä p2 Ä Ä ps , then r D s and qi D pi for all i . We do this by induction on n. First check the case n D 1; 1 cannot be written as the product of any nonempty collection of prime numbers. So consider a natural number n 2 and assume inductively that the assertion of unique factorization holds for all natural numbers less than n.

For each invertible map f W X ! Y , there is a map f 1 W Y ! X called the inverse of f and satisfying f ı f 1 D idY and f 1 ı f D idX . Here idX denotes the identity map idX W x 7! x on X and similarly idY denotes the identity map on Y . x/ D y. Now consider maps from a set X to itself. Maps from X to itself can be composed, and composition of maps is associative. If f and g are bijective maps on X , then the composition f ı g is also bijective (with inverse g 1 ı f 1 ). X/ satisfies the following properties: 1.

Explain how to compute the inverse of a permutation that is given in two-line notation. 7. Explain how to compute the inverse of a permutation that is given as a product of cycles (disjoint or not). One trick of problem solving is to simplify the problem by considering special cases. First you should consider the case of a single cycle, and it will probably be helpful to begin with a short cycle. A 2-cycle is its own inverse, so the first interesting case is that of a 3-cycle. Once you have figured out the inverse for a 3-cycle and a 4-cycle, you will probably be able to guess the general pattern.