Number theory, geometry and algebra - Dynamics-approx.jku.at

Z(g)hasφ(r)generators, namelythoseelementsg s 1
,...,g s φ(r) ,wheres1 ,...,s φ(r)
is a listing of Z ∗ m (one calls it a reduced residue class for Z r).
Of course, if the group is finite, the first case cannot occur and the order
of g as well as b eing finite is a divisor of |G|. (This provides a new proof
for the theorem of Fermat: a φ(m) = 1 (modm), if m ∈ N, m > 1, and a ∈ Z
with gcd(a,m) = 1).
Remark 1) If ord g = r = kl where k,l > 0, then ord(g k ) = l.
2) If ord g 1 and ord g 2 are relatively prime, and the group is commutative,
then ord g 1 g 2 = (ord g 1 )(ord g 2 ).
Proof. Put r = ord g 1 , s = ord g 2 . We have
(g 1 g 2 ) rs = g rs
1 g rs
2 = e.
Hence t|rs where t = ord g 1 g 2 . Since (g 1 g 2 ) rt = e = g1 rt , then s|rt, and so s|t.
Similarlyl, r|t and so rs|t. This imlpies t = rs.)
We shall now show that the arithmetical function φ introduced above is
multiplicative.
Proposition 17 Let {r 1 ,...,r φ(m) } be a reduced system of residues (mod
m), resp. {s 1 ,...,s φ(n) } a reduced system of residues (mod n), whereby m
and n are relatively prime. Then the numbers
{nr i +ms j : i ∈ {1,...φ(m)}, j ∈ {1,...,φ(n)}}
are a reduced system (mod mn).
Corollar 3 φ(mn) = φ(m)φ(n) provided that gcd(m,n) = 1.
We bring an alternative proof of the following result:
∑
φ(d) = n (n ∈ N)
d|n
Let D(d) denote the number of y from {1,...,n}, with gcd(n,y) = d. Since
every number is associated to a D(d),
n = ∑ d|n
|D(d)|.
But |D(d)| = φ( n). For x ∈ D(d) ⇔ x = qd where 1 ≤ q ≤ n and
d d
gcd( n,q) = 1. There are exactly d φ(n ) such q’s.
d
We now consider the following problem: When is the group Z ∗ n cyclic?
Thisisequivalent tothequestion: Doesthereexistg ∈ Z ∗ n withordg = φ(n)?
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