External Direct Products (11/4)
• Definition. If G1, G2, ..., Gn are groups, then their external
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direct product G1 G2 ... Gn is simply the set of all
ordered n-tuples of elements of the groups under
component-wise operations.
Example: What are the elements of Z3 Z4? How many?
What is the sum of (1, 2) and (2, 3)? (We say “sum
because these are additive group.)
What is the order of (1, 1)? What does this say about this
direct product group?
Example: What are the elements of Z2 Z6? How many?
What is the order of (1, 1) in this group? Is it cyclic?
Basic Results
• Theorem (Order of the Product Group). If all groups in an
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external product are finite, then the order of the product
group is just ______________________________.
This is because as a set, the external product is just the
“cartesian product” from set theory.
Theorem (Order of an element in a Product Group). The
order the element (a1, a2,..., an) is _______________
___________________________________________.
Corollary. The group Zj Zk is cyclic if and only if
_________________________________.
Generalize to n components.
Examples
• In each case below
1. How many elements does the group have?
2. What is the largest order of any element in the group?
Hence, is it cyclic?
3. Write each as being isomorphic to another product with
as few factors as possible.
• Z2
Z9 Z5
• Z2
Z3 Z3 Z5
• Z4
Z3 Z9 Z5 Z5
Groups of Order 4
• Theorem. Every group G of order 4 is isomorphic to either
Z4 (i.e., is cyclic) or Z2 Z2 (i.e., is a Klein-4 group).
• Proof. If G has an element of order 4, then G is cyclic,
hence isomorphic to Z4. Otherwise G must consist of
{e, a, b, ab}, the latter 3 of order 2 (and, recall, ab=ba).
Now the function taking e to (0, 0), a to (1, 0), b to (0, 1)
and ab to (1, 1) is easily checked to be an isomorphism.
• This is an example of the upcoming Fundamental
Theorem of Finite Abelian Groups, which says that every
finite abelian group can be expressed as a direct product
of cyclic, i.e., Zn, groups.
The Structure of the U groups
• Theorem. If s and t are relatively prime, then U(st) is
isomorphic to U(s) U(t).
• Theorem (Gauss).
U(2) {0},
U(4) Z2,
for n > 2, U(2n) Z2 Z2^(n – 2), (i.e., is not cyclic), and
for p any odd prime, U(pn) Zp^n - p^(n – 1)
(i.e., is cyclic).
• Examples. Check this for n = 8, 9, 10, 12, 16.
Assignment for Wednesday
• Hand-in #3 is due.
• Read Chapter 8 through page 168.
• On page 174, do Exercises 2, 5, 6, 8, 9, 10, 12, 13, 14,
15.