Algebra I: Section 6. The structure of groups. 6.1 Direct products of ...

6.1.32 Theorem. If m, n > 1 are relatively prime and Um, Un, Umn are the multiplicative
abelian groups of units in Zm, Zm, Zmn then
(8) (Umn, · ) ∼ = (Um, · ) × (Un, · )
and in particular the sizes of these groups satisfy the following multiplicative condition
(9) |Umn| = |Um| · |Un| if gcd(m, n) = 1
Note: Before launching into the proof we remark that the set of units in any commutative ring
R with identity 1 R ∈ R can be defined just as for Zn
(10) UR = {x ∈ R : ∃ y ∈ R such that x · y = 1 R }
The element y in (10) is called the multiplicative inverse of x, and is written y = x −1 . As with
Zn, the units UR form a commutative group (UR, · ) under multiplication. The proof of 6.1.32
rests on two easily verified properties of groups UR.
6.1.33 Exercise. If ψ : R → R ′ is an isomorphism of commutative rings with identity show
that ψ must map units to units, so that ψ(UR) = UR ′. In particular the groups (UR, · ) and
(UR ′, · ) are isomorphic. While you are at it, explain why ψ must map the identity 1 ∈ R to
R
the identity 1R ′ ∈ R′ . �
6.1.34 Exercise. If R1, R2 are commutative rings with identities 11 ∈ R1, 12 ∈ R2, prove that
the set of units UR1×R2 in the direct product ring is the Cartesian product of the separate
groups of units
(11) UR1×R2 = UR1 × UR2 = {(x, y) : x ∈ UR1, y ∈ UR2} �
Proof (6.1.32): First observe that ψ : Zmn → Zm × Zn maps Umn one-to-one onto the group
of units UZm×Zn, by 6.1.33. By 6.1.43, the group of units in Zm × Zn is equal to Um × Un.
Since ψ is a bijection that intertwines the (·) operations on each side we see that ψ is a
group isomorphism from (Umn, · ) to the direct product group (Um, · ) × (Un, · ). �
Remarks: Even if m and n are relatively prime, the orders |Um|, |Un| of the groups of units
need not have this property. For instance, if p > 1 is a prime then |Up| = p − 1 and there
is no simple connection between p and the prime divisors of p − 1, or between the divisors of
|Up| = p − 1 and |Uq| = q − 1 if p, q are different primes. Furthermore the groups Um, Un need
not be cyclic, though they are abelian. Nevertheless we get a direct product decomposition (8).
Theorem 6.1.32 operates in a very different environment from that of the Chinese Remainder
Theorem 6.1.26. Its proof is also more subtle in that it rests on the interplay between the (+)
and (·) operations in Z, while the CRT spoke only of (+).
6.1.35 Exercise. If n1, . . . , nr are pairwise relatively prime, with gcd(ni, nj) = 1 for i �= j,
give an inductive proof that ∼= Un1n2...nr Un1 × . . . × Unr and |Un1n2...nr| = |Un1| · . . . · |Unr| .
�
6.1.36 Exercise. Decompose Z630 as a direct product Zn1 × . . . Znr of cyclic groups. Using
2.5.30 we could compute |U630| by tediously comparing the prime divisors of 630 = 2 · 3 2 · 5 · 7
with those of each 1 ≤ k < 630. However, the formula of 6.1.35 might provide an easier way to
calculate |U630|. Do it.
Hint: Is the group of units (U9, · ) cyclic? (Check orders o(x) of its elements.) �
The Euler Phi-Function φ : N → N is often mentioned in number theory. It has many
equivalent definitions, but for us it is easiest to take
φ(1) = 1 φ(n) = |Un| = #(multiplicative units in Zn)
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