Math 3121 Abstract Algebra I

Math 3121Abstract Algebra I Lecture 9 Finish Section 10 Section 11

HW Due Today • Hand in: Due Tues, Oct 28:Page 73: 12, 14, 16Page 84: 18 Pages 94-95: 10, 24, 36 • Do not hand in: Pages 94-95: 19, 39

Section 10 • Section 10: Cosets and the Theorem of Lagrange • Modular relations for a subgroup • Definition: Coset • Theorem of Lagrange: For finite groups, the order of subgroup divides the order of the group. • Theorem: For finite groups, the order of any element divides the order of the group

Modulo a Subgroup Definition: Let H be a subgroup of a group G. Define relations: ~L and ~R by: x ~L y ⇔ x-1 y in H x ~R y ⇔ x y-1 in H We will show that ~L and ~R are equivalence relations on G. We call ~L left modulo H. We call ~R right modulo H. • Note: x ~L y ⇔ x-1 y = h, for some h in H ⇔ y = x h, for some h in H x ~R y ⇔ x y-1 = h, for some h in H ⇔ x = h y, for some h in H

Modulo a Subgroup is an Equivalence Relation Theorem: Let H be a subgroup of a group G. The relations: ~L and ~R defined by: x ~L y ⇔ x-1 y in H x ~R y ⇔ x y-1 in H are equivalence relations on G. Proof: We show the three properties for equivalence relations: 1) Reflexive: x-1 x = e is in H. Thus x ~L x. 2) Symmetric: x ~L y ⇒x-1 y in H ⇒ (x-1 y) -1 in H ⇒y-1 x in H ⇒y ~L x 3) Transitive: x ~L y and y ~L z ⇒x-1 y in H and y-1 z in H ⇒ (x-1 y )( y-1 z) in H ⇒ (x-1 z) in H ⇒x ~L z Similarly, for x ~R y .

Cosets • The equivalence classes for these equivalence relations are called left and right cosets modulo the subgroup. Recall: x ~L y ⇔ x-1 y = h, for some h in H ⇔ y = x h, for some h in H • Cosets are defined as follows Definition: Let H be a subgroup of a group G. The subset a H = { a h | h in H } is called the left coset of H containing a, and the subset H a= { a h | h in H } is called the right coset of H containing a.

Examples • Cosets of nℤ in ℤ are: nℤ, nℤ+1, nℤ+2, …, nℤ + (n-1) • For example 2ℤ in ℤ has two cosets. • Cosets of {0, 3} in ℤ6 • Cosets of {0, 2, 4} in ℤ6

Abelian versus Nonabelian • Note: In abelian groups the left cosets are the right cosets. In nonabelian case: left and right don’t always agree. • H = {e, μ1} in G = S3 has different left and right cosets. For left cosets, make a multiplication table G×H. For right cosets, make a multiplication table H×G. (See slides after this one). • H = {ρ0, ρ1, ρ2} in S3 has same left and right cosets.

Multiplication Table for S3 in Cycle Notation

Left Cosets for S3 in Cycle Notation Distinct left cosets of H = {e, (2 3)} {e, (2 3)} = e H = (2 3) H {(1 2 3), (1 2)} = (1 2 3) H = (1 2) H {(1 3 2), (1 3)} = (1 3 2) H = (1 3) H

Right Cosets of <(2 3)> for S3 in Cycle Notation Distinct right cosets of H = {e, (2 3)} {e, (2 3)} = H = H (2 3) {(1 2 3), (1 3)} = H (1 2 3) = H (1 3) {(1 3 2), (1 2)} = H (1 3 2) = H (1 2)

Counting Cosets Theorem: For a given subgroup of a group, every coset has exactly the same number of elements, namely the order of the subgroup. Proof: Let H be a subgroup of a group G. Recall the definitions of the cosets: aH and Ha. a H = { a h | h in H } H a= { a h | h in H } Define a map La from H to aH by the formula La(g) = a g. This is 1-1 and onto. Define a map Ra from H to Ha by the formula Ra(g) = g a. This is 1-1 and onto.

Lagrange Theorem (Lagrange): Let H be a subgroup of a finite group G. Then the order of H divides the order of G. Proof: Let n = number of left cosets of H, and let m = the number of elements in H. Then m is the number of elements in any left coset. Thus n m = the number of elements of G. Here m is the order of H, and n m is the order of G.

Orders of Cycles • The order of an element in a finite group is the order of the cyclic group it generates. Thus the order of any element divides the order of the group.

HW Section 10 • Don’t hand in: • Pages 101: 3, 6, 9, 15 • Hand in Tues, Nov 4 • Pages 101-102: 6, 8, 10, 36, 40

Section 11 (as time permits) • Direct Products and Finitely Generated Abelian Groups • Cartesian Product of sets • Direct product of groups • Structure of ZnZm • Structure of products of cyclic groups • Next time: Structure of Finitely Generated Abelian Groups

Cartesian Products Definition: The Cartesian product of a finite collection of sets Sk, for k = 1 to n is the set of all n-tuples (s1, s2, …, sn), with sk in Sk. The Cartesian product is denoted by S1 ×S2 × … ×Sn or by product notations such as

Projection Maps • The map pi: S1 ×S2 × … ×Sn Si that takes the n-tuple s = (s1, s2, …, sn) to its ith component si is called the ith projection map. • In other words: For any s in S1 ×S2 × … ×Sn the result of applying the ith projection map to s is called the ith component of s and denoted by si . • Note: Two elements a and b of the product are equal if and only if ai = bi, for all i = 1,…,n.

Direct Product of Groups Theorem: Let G1, G2, …, Gn be groups with multiplicative notation. Define a binary operation on the Cartesian product G1×G2×… ×Gn by (a1, a2, …, an)(b1, b2, …, bn) = (a1 b1, a2 b2, …, an bn), then G1×G2×… ×Gn is a group with this binary operation. The set G1×G2×… ×Gn with this binary operation is called the direct product of the groups G1, G2, …, Gn. Proof: Note that the binary operation is defined component-wise. That is, if a and b are in the product, then (a b)i = ai bi. 1) Associativity follows because each component binary operation is associative. 2) The identity is e the n-tuple e = (e1, e2, …, en), where each ei is the identity of its own component group Gi. 3) For each a = (a1, a2, …, an) in the product, the n-tuple a-1 = ((a1)-1, (a2)-1, …, (an)-1) is the inverse of a.

Direct Sums of Groups • Sometimes we call the direct product a direct sum, especially if we use additive notation. • A direct product is characterized by its projection maps. These turn out to be homomorphisms. • On the other hand, the direct sum is characterized by injection maps ji: Gi G1×G2×… ×Gn that each take ai in Gi to the n-tuple (e1, e2, …, ai, …, en) that has identities in all components except for the ith, which has ai. These also turn out to be homomorphisms.

Examples • ℤ2×ℤ3 • ℤ3×ℤ5 • ℤ2×ℤ2 • ℤ3×ℤ3 • ℤ2×ℤ6 • ℤ9×ℤ6

Internal Products • Each component group of a direct product can also be considered a subgroup by injection. All of these subgroups commute with each other. Thus any element g of the product G1×G2×… ×Gn can be uniquely written in the form g = g1 g2 … gn with gi in Gi. When this happens, G is said to be an internal product of the subgroups Gi.

LCM and GCD of two numbers • Let x and y be integers, then • LCM(x, y) is the least multiple of x and y: LCM(x, y) = min{ m in ℤ+ | m is a multiple of x and m is a multiple of y} • GCD(x, y) is the greatest common divisor of x and y: GCD(x, y) = max{ m in ℤ+ | m divides x and m divides y} • LCM(x, y) = x y /GCD(x, y).

Methods of finding LCM and GCD • Euclidean Algorithm to find GCD in the form a x + b y: Start with positive integers x and y. Set r0 = x, r1 = y. Given rk-1 and rk, find qk and rk+1 such that: rk-1 = qk rk + rk+1, with 0 ≤ rk+1 < rk. Continue until rk+1 = 0. Then GCD(x, y) = rk. Then work backward to write GCD in the form a x + b y. • Example: Find GCD of 64 and 58. • For small numbers use prime factorization.

Math 3121 Abstract Algebra I