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 ℤ n × ℤ m is cyclic and is isomorphic to the group ℤ n m if and only if n is relatively prime to m. Proof: For r in ℤ n and s in ℤ m, 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 ℤ n × ℤ m. Thus (1, 1) generates the group and thus ℤ n × ℤ m is 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 ℤ n × ℤ m. Thus (r, s) cannot generate it. Thus ℤ n × ℤ m is not cyclic.

LCM of a finite number of numbers Definition: Let r 1, r 2, …, r n be positive integers. The least common multiple of r 1, r 2, …, r n is the least positive integer that is a multiple of all of them. This is denoted by LCM(r 1, r 2, …, r n ). More formally: LCM(r 1, r 2, …, r n ) = min{m in ℤ + | m is a multiple of r i, for all i = 1, …, n}.

Order of a member of the product Theorem: Let a be in G 1 ×G 2 ×… ×G n Let r i be the order of the ith component of a. Then the order of a is LCM(r 1, r 2, …, r n ). Proof: a m = e if and only if (a m ) i = e i, for all i = 1, … n. Each (a m ) i = a i m. Thus a m = e if and only if a i m = e i, for all i = 1, …, n. Thus, m is a multiple of the order of a if and only m is a multiple of r i, for all i = 1, …, n. The result follows.

Example Find the order of (10, 8, 16) in ℤ 24 × ℤ 12 × ℤ 18

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 ℤ n. 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 = Primary form: ℤ 5 × ℤ 5 × ℤ 2 × ℤ 2 ℤ 25 × ℤ 2 × ℤ 2 ℤ 5 × ℤ 5 × ℤ 4 ℤ 25 × ℤ 4 What about 2) the division form?

HW Don’t hand in – pages : 1, 3, 5, 7, 9, 15, 17, 25, 29, 39 Hand in (Due Nov 4): – page : 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  Z n The exponential map for real and complex numbers Parity: from S n to Z 2 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).