This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Notes: c circlecopyrt F.P. Greenleaf 2003  2010 v43f10products.tex, version 12/1/10 Algebra I: Chapter 6. The structure of groups. 6.1 Direct products of groups. We begin with a basic product construction. 6.1.1 Definition (External Direct Product). Given groups A 1 ,...,A n we define their external direct product to be the Cartesian product set G = A 1 × ... × A n equipped with componentbycomponent multiplication of ntuples. If a = ( a 1 ,...,a n ) , b = ( b 1 ,...,b n ) in the Cartesian product set G , their product is (1) a · b = ( a 1 ,...,a n ) · ( b 1 ,...,b n ) = ( a 1 b 1 ,...,a n b n ) for all a i ,b i ∈ A i The identity element is e = ( e 1 ,...,e n ) where e i is the identity element in A i ; the inverse of an element is a − 1 = ( a − 1 1 ,...,a − 1 n ) . There is a natural isomorphism between A i and the subgroup A i = ( e 1 ) × ... × A i × ... ( e n ) , the ntuples whose entries are trivial except for a i . From (1) it is clear that (a) Each A i is a subgroup in G . (b) The bijective map J i ( a i ) = ( e 1 ,...,a i ,...,e n ) defines an isomorphism from A i to A i . (c) The A i commute with each other in the sense that xy = yx if x ∈ A i , y ∈ A j and i negationslash = j . (d) Each A i is a normal subgroup in G . (e) The product set A 1 · ... · A n = { x 1 ...x n : x i ∈ A i , 1 ≤ i ≤ n } is all of G . Note carefully what (c) does not say: the subgroup A i need not commute with itself (the case when i = j ) unless the group A i happens to be abelian. The subsets H i = A 1 × ... × A i − 1 × ( e i ) × A i +1 × ... × A n ⊆ G are also normal subgroups, and in a grouptheoretic sense the H i are complementary to the A i . We have the following properties. (2) (i) H i ∩ A i = ( e ) (ii) G = A 1 · A 2 · ... · A n (product of subsets in G ) (iii) Each complement H i is a normal subgroup in G (iv) A i ∼ = G/ H i via the bijection f i : a i mapsto→ J i ( a i ) H i 6.1.2 Exercise. Verify the claims (a) – (e) regarding the subgroups A i in a direct product G = A 1 × ... × A n . square 6.1.3 Exercise. Verify the relations (2) between the subgroups A i ∼ = A i and their complementary subgroups H i . square 6.1.4 Exercise. Verify that the map f i : A i → G/ H i defined in (iii) above is actually a 1 bijection, and that it is a homomorphism from A i to the quotient group G/ H i , so that G/ H i ∼ = A i . square The order of entries in an ntuple makes a difference; therefore the Cartesian product sets A 1 × A 2 and A 2 × A 1 are not the same thing (unless A 1 = A 2 ). For instance, are the direct product groups Z 3 × Z 5 and Z 5 × Z 3 the same ? What do elements in these groups look like? However, in dealing with groups we only care whether they are isomorphic. It happens that Z 3 × Z 5 ∼ = Z 5 × Z 3 even though these groups are not “identical.” 6.1.5 Exercise. Let A 1 ,A 2 ,... be groups. Prove that the following product groups are isomorphic....
View
Full
Document
This note was uploaded on 03/25/2011 for the course MATH 301 taught by Professor Algebra during the Spring '11 term at NYU.
 Spring '11
 ALgebra
 Algebra

Click to edit the document details