206
3. PRODUCTS OF GROUPS
The elementary divisors of a finite abelian group can be obtained from
any direct product decomposition of the group with cyclic factors, as illus
trated in the previous example. If
G
Š
Z
a
1
Z
a
n
, then the elementary
divisors are the prime power factors of the integers
a
1
; a
2
; : : : ; a
n
.
The invariant factors can be obtained from the elementary divisors by
the following algorithm, which was illustrated in the example:
1.
Group together the elementary divisors belonging to each prime
dividing the order of
G
, and arrange the list for each prime in
weakly decreasing order.
2.
Multiply the largest entries of each list to obtain the largest in
variant factor.
3.
Remove the largest entry in each list.
Multiply the largest re
maining entries of each nonempty list to obtain the next largest
invariant factor.
4.
Repeat the previous step until all the lists are exhausted.
Example 3.6.24.
In the previous example, the lists of elementary divisors,
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 EVERAGE
 Algebra, Factors, Prime number, Cyclic group, Zan, largest invariant factor, elementary divisors

Click to edit the document details