226 CHAPTER 4 Abstract Algebra
Z∗p for infinitely many primes p. This is often called theArtin
Conjecture, and the answer is “yes” if one knows that the so-
calledGeneralized Riemann Hypothesisis true! Try checking
this out for the first few primes, noting (as above) that 2 isnota
generator forZ∗ 7.- Letnbe a positive integer and consider the set of complex numbers
Cn={e^2 πki/n= cos 2πk/n+isin 2πk/n|k= 0, 1 , 2 , ..., n− 1 }⊆C.
If we set ζ = e^2 π/n, then e^2 πki/n = ζk. Since also ζn = 1 and
ζ−^1 =ζn−^1 we see that not only isCnclosed under multiplication,
it is in fact, a cyclic group.We hasten to warn the reader that in a cyclic group the generator is
almost never unique. Indeed, the inverse of any generator is certainly
also a generator, but there can be even more. For example, it is easy
to check that every non-identity element of the additive cyclic (Z 5 ,+)
is a generator. This follows by noting that 1 is a generator and that
1 = 3·2 = 2·3 = 4· 4.On the other hand, we showed above that 3 is a generator of the cyclic
groupZ∗ 7 , and since 3−^1 = 5 (because 3·5 = 1), we see that 5 is also
a generator. However, these can be shown to be the only generators of
Z∗ 7. In general, ifG is a cyclic group of ordern, then the number of
generators ofGisφ(n), where, as usual,φis the Eulerφ-function; see
Exercise 6, below.
In fact, we’ll see in the next section that if (G,∗) is a group of prime
orderp, then not only isGcyclic, every non-identity element ofGis a
generator.
We shall conclude this section with a useful definition. Let (G,∗) be
a group, and letg∈G. Theorderofgis the least positive integern
such thatgn=e. We denote this integer byo(g). If no such integer ex-
ists, we say thatghasinfinite order, and writeo(g) =∞. Therefore,
for example, in the groupZ∗ 7 , we have