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If n is a positive integer and a is relatively prime In order to apply Corollary 2 one should replace a by its remainder on division by n. If n should be a prime number p, then cp(p) = p - 1. If a is an integer relatively prime to p, then by Corollary 3, ap- 1 1 mod p, Whence aP a mod p. If, on the other hand, a is not relatively prime top, = = 43 44 Group Theory Ch. 2 since p is a prime number, we must have that hence 0 aP a mod p here also. Thus = = If pis a prime COROLLARY 4 (FERMAT) aP = a modp.

The reader may verify that this is a group. It is called a cyclic group of order n. 4 may be achieved as follows: Let S be the circle, in the plane, of radius 1, and let p n be a rotation through an angle of2njn. Then Pn E A(S) and Pn in A(S) generates a group of order n, namely, {e, Pm Pn 2 , . . , Pnn- 1 }. 5 Let S be the set of integers and, as usual, let A(S) be the set of all one-to-one mappings of S onto itself. Let G be the set of all elements in A(S) which move only a finite number of elements of S; that is, u E G if and only if the number of x in S such that xu =1= x is finite.

8. If the set S has a finite number of elements, prove the following: (a) If a maps S onto S, then a is one-to-one. (b) If a is a one-to-one mapping of S onto itself, then u is onto. (c) Prove, by example, that both part (a) and part (b) are false if S does not have a finite number of elements. 9. 2 are false; namely, (a) If a o-r is onto, it need not be that both u and -r are onto. (b) If a o -r is one-to-one, it need not be that both u and -r are one-toone. 10. Prove that there is a one-to-one corresponden ce between the set of integers and the set of rational numbers.

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