We have the basic tools required to studied the

structure of groups through their subgroups and

their individual elements and by means of

isomorphisms between groups. This allows us to

build up larger groups from smaller ones.

Let
be groups. Then the** external
direct product **of these groups, denoted

or , is the set of all n-tuples

for each i = 1, 2, … , n}.

Combining elements of this set under the

componentwise operations of the G

external direct product a group. (Why?)

Examples:

• We are already very familiar with vector

addition in the groups

• is a

cyclic group, generated by (1,1). The map from

to defined by k(1,1) k is an

isomorphism between them (why?).

• is not cyclic

(why?). Thus, and are structurally

distinct groups.

**Theorem** The order of
is

**Proof** The order of is the smallest

exponent k for which

equals . Thus,
for each i. So k is

a multiple of for each i, and being the
smallest

number with this property forces k to be exactly

//

**Theorem** If G and H are finite cyclic groups, then

G ⊕H is cyclic |G| and |H| are relatively

prime.

**
Proof **If |G|= m and |H|= n, then |G ⊕H|= mn.

So if G ⊕H is cyclic and has (g, h) as a generator,

then with d = gcd(m, n), we have

But then . So m and n
are

relatively prime. Conversely, if m and n are

relatively prime, then l
(g,h) l= lcm(m, n) = mn,

whence (g, h) must be a generator for
G ⊕H. //

**Corollary** If
are finite cyclic
groups,

then is cyclic
the orders l G_{i}
l are
relatively

prime in pairs. //

**Corollary**
the n_{i}’s

are relatively prime in pairs. //

(This last corollary is equivalent to an important

result in number theory which is known as the

Chinese Remainder Theorem and is often expressed

in terms of congruence arithmetic.)

**Theorem** If s and t are relatively prime, then

. In fact, the subgroup

of U(st) is isomorphic to U(s), and similarly, the

subgroup .

**Proof **We claim that the map from U(st) to

given by (x mod s ,x mod t) is an

isomorphism. This follows from the claim that the

map from to U(s) given by
x x mod s is an

isomorphism. The details are omitted here. //

**Corollary **If the prime factorization of the integer

n is (where the numbers p_{i} are distinct

prime numbers), then . //

Gauss, in a work titled Disquisitiones Arithmeticae,

a monumental work in number theory published in

1801, proved results which we can interpret today

in the form of the following

**Theorem**
U(2) is trivial; ; and for n > 2,

. Moreover, for any odd prime p,

From Gauss’ theorem, it is possible to realize any

group of units U(n) as an external direct sum of

cyclic groups.

Example:

How many elements of order 12 are there in

U(720)? The same number as there are in

. Viewing such an element in

the form (a, b, c, d), we note that this question now

takes the form

where

l a l= 1 or 2

l b l= 1, 2, or 4

l c l= 1, 2, 3, or 6

l d l= 1, 2, or 4

Given these options,
c must equal 3 or 6 and one of

l
b l or l
d l (or both) must equal 4. This gives three

cases:

(1) l
a l= 1 or 2, l
b l= 4, l
c l = 3 or 6, l
d l= 1 or 2;

(2) l
a l= 1 or 2, l
b l= 1 or 2, l
c l= 3 or 6, l
d l= 4;

(3) l
a l= 1 or 2, l
b l= 4, l
c l= 3 or 6, l
d l= 4.

Recalling that there are exactly
φ (d) elements of

order d in a cyclic group, we can count the number

of elements of given orders in each of these

possibilities: there will be one element of orders 1

and 2, two elements of orders 3, 4 and 6. Putting

all this together gives in each case:

(1) (1 + 1)(2)(2 + 2)(1 + 1) = 32 elements;

(2) (1 + 1)(1 + 1)(2 + 2)(2) = 32 elements;

(3) (1 + 1)(2)(2 + 2)(2) = 32 elements;

for a total of 96 elements of order 12 in U(720).