The Group of Units

Abstract

We saw in Chapter 5 that for each n, the set U _n of units in ℤ_n forms a group under multiplication. Our aim in this chapter is to understand more about multiplication and division in ℤ_n by studying the structure of this group. An important result is that if n = p ^e, where p is an odd prime, then U _n is cyclic; following a commonly-used strategy, we shall prove this first for n = p, and then deduce it for n = p ^e. As often happens in number theory, the prime 2 is exceptional: although U ₂ and U ₄ are cyclic, we shall see that the group U ₂ ^e is not cyclic for e ≥ 3, although in a certain sense it is nearly cyclic. Using the Chinese Remainder Theorem, we can use our knowledge of the prime-power case to deduce the structure of U _n for arbitrary n. As an application, we will continue the study of Carmichael numbers, begun in Chapter 4.