Download Math 3121 Lecture 10

Math 3121
Abstract Algebra I
Lecture 10
Finish Section 11
Skip 12 – read on your own
Start Section 13
When is a direct product of cyclic
groups cyclic?
Theorem: The group ℤn×ℤm is cyclic and is isomorphic to the group ℤn m if
and only if n is relatively prime to m.
Proof: For r in ℤn and s in ℤm, the subgroup generated by any element (r, s)
has order equal to the least common multiple of the order of r and the
order of s. Let order(r) = x = n/GCD(r, n) and y = order(s) = m/GCD(s, n).
Then order((r,s)) = LCM(x, y) = x y /GCD(x, y).
If GCD(n, m) =1, then r=1 and s=1 make the order of (r, s) equal to m n
which is the order of ℤn×ℤm. Thus (1, 1) generates the group and thus
ℤn×ℤmis cyclic.
Conversely, if GCD(n, m) = d >1, then m n/d is divisible by both m and n.
Hence (m n/ d)(r, s) = 0. Thus the order of ( r, s) is less than the order of
ℤn×ℤm. Thus (r, s) cannot generate it. Thus ℤn×ℤm is not cyclic.
LCM of a finite number of numbers
Definition: Let r1, r2, …, rn be positive integers.
The least common multiple of r1, r2, …, rn is
the least positive integer that is a multiple of
all of them. This is denoted by LCM(r1, r2, …,
rn). More formally:
LCM(r1, r2, …, rn) = min{m in ℤ+ | m is a multiple
of ri, for all i = 1, …, n}.
Order of a member of the product
Theorem: Let a be in G1×G2×… ×Gn Let ri be the
order of the ith component of a. Then the
order of a is LCM(r1, r2, …, rn).
Proof: am = e if and only if (am)i = ei , for all i = 1,
… n. Each (am)i = aim. Thus am = e if and only if
aim = ei , for all i = 1, …, n. Thus, m is a
multiple of the order of a if and only m is a
multiple of ri, for all i = 1, …, n. The result
follows.
Example
• Find the order of (10, 8, 16) in ℤ24×ℤ12×ℤ18
Classification Theorems for Finitely
Generated Abelian Groups
Theorem: Any finitely generated abelian group is
isomorphic to a direct product of so many copies of
ℤ and ℤn.
ℤ n[1] × ℤ n[2] × … × ℤ n[r] × ℤ × … × ℤ
There are two standard forms:
1) Each n[i] is a power of some prime p[i]. The
primes p[i] are not necessarily different.
2) n[i] is divisible by n[j], for j > i, n[r] >=2.
Proof: The proof is beyond the scope of the course.
Examples
• Subgroups of order 100:
100 = 22 52
Primary form:
ℤ5 × ℤ5 × ℤ2 × ℤ2
ℤ25 × ℤ2 × ℤ2
ℤ5 × ℤ5 × ℤ4
ℤ25 × ℤ4
What about 2) the division form?
HW
• Don’t hand in
– pages 110-113: 1, 3, 5, 7, 9, 15, 17, 25, 29, 39
• Hand in (Due Nov 4):
– page 110-113: 10, 12, 16, 22, 24
Section 12
• Read this.
Section 13
• Homomorphisms
– Definition of homomorphism (recall)
– Examples
– Properties
– Kernel and Image
– Cosets and inverse images
– 1-1
– Normal Subgroups
Definition of Homomorphism
Definition: A map f of a group G into an group
G’ is called a homomorphism if it has the
homomorphism property:
f(x y ) = f(x) f(y), for all x, y in G
Note: This definition uses multiplicative
notation. Recall what happens when we use
formal notation or we switch between
additive and multiplicative notation.
Examples of Homomorphisms
•
Multiplication in Z:
f: Z  Z
x ↦ nx
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The canonical map: Z  Zn
The exponential map for real and complex numbers
Parity: from Sn to Z2
Projections from a direct product
Evaluating real valued functions at a point. (Pick any set X, and consider the
functions from X to the real numbers).
Taking integrals of continuous functions.
Evaluating group-valued functions on a set at a point in that set. (Pick any set X,
and consider the functions from X to a group G. Pick any point x in X.)
The determinant of matrices in GL(n,R).