Math 3121Abstract Algebra I

Lecture 10Finish Section 11

Skip 12 – read on your ownStart 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 = ai

m. Thus am = e if and only if ai

m = 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 × … × ℤThere 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 × ℤ4ℤ25 × ℤ4What 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

• 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).