v43-f10groupsfinal

v43-f10groupsfinal - Notes F.P Greenleaf c circlecopyrt...

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Unformatted text preview: Notes: F.P. Greenleaf, c circlecopyrt 2000-2010 v43-f10groups.tex, version 10/18/10 Algebra I: Chapter 3. Group Theory 3.1 Groups. A group is a set G equipped with a binary operation mapping G × G → G . Such a “product operation” carries each ordered pair ( x,y ) in the Cartesian product set G × G to a group element which we write as x · y , or simply xy . The product operation is required to have the following properties. G.1 Associativity: ( xy ) z = x ( yz ) for all x,y,z ∈ G . This insures that we can make sense of a product x 1 ··· x n involving several group ele- ments without inserting parentheses to indicate how elements are to be combined two at a time. However, the order in which elements appear in a product is crucial! While it is true that x ( yz ) = xyz = ( xy ) z , the product xyz can differ from xzy . G.2 Unit element: There exists an element e ∈ G such that ex = x = xe for all x ∈ G . G.3 Inverses exist: For each x ∈ G there exists an element y ∈ G such that xy = e = yx . The inverse element y = y ( x ) in G.3 is called the multiplicative inverse of x , and is generally denoted by x − 1 . The group G is said to be commutative or abelian if the additional axiom G.4 Commutativity: xy = yx for all x,y ∈ G is satisfied Our first task is to show that the identity element and multiplicative inverses are uniquely defined, as our notation suggests. 3.1.1 Lemma. In a group ( G, · ) the unit e is unique, and so is x − 1 for each x . P roof : Suppose there is another element e ′ ∈ G such that e ′ x = x = xe ′ for all x ∈ G . Taking x = e we get e ′ = e ′ e = e as claimed. Next, let x ∈ G and suppose y,y ′ are elements such that xy ′ = e = y ′ x, xy = e = yx . Then look at the product y ′ xy and apply G.1+G.2 to get y ′ = y ′ e = y ′ ( xy ) = ( y ′ x ) y = ey = y Thus y ′ = y and every x has a unique inverse which we hereafter label x − 1 . square 3.1.2 Some examples of groups. We write | G | for the number of elements in G , which could be ∞ . 1. G = { e } . This is the trivial group with just one element e such that e · e = e . Here e − 1 = e and | G | = 1. Not very interesting. square 2. G = ( Z , +). This is an infinite abelian group; integer addition (+) is the group operation. The unit is e = 0, and the inverse of any element x ∈ Z is its negative − x . square 3. G = ( Z n , +), the integers (mod n ) for some n ∈ N , with addition as the group operation. This is a finite abelian group with | G | = n . The identity element is [0]; the inverse of [ k ] ∈ Z n is the congruence class [ − k ] = [ n − k ]. square 4. G = (U n , · ), the set of multiplicative units in Z n . Here we take multiplication [ k ] · [ ℓ ] = [ kℓ ] as the group operation. Recall that U n can also be described as U n = { [ k ] : 0 < k < n and gcd( k,n ) = 1 } 1 as explained in 2.5.15. You should also recall the discussion of Section 2.5, where ( Z n , + , · ) was defined, to see why the group axioms are satisfied. The) was defined, to see why the group axioms are satisfied....
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This note was uploaded on 03/25/2011 for the course MATH 301 taught by Professor Algebra during the Spring '11 term at NYU.

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v43-f10groupsfinal - Notes F.P Greenleaf c circlecopyrt...

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