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# Note for CSCI 124 with Professor Vora at GW

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Date Created: 02/07/15

CSCI 124224 Discrete Structures 11 Homomorphisms and Isomorphisms PoorVi L Vora In this module we look at functions on groups that preserve structure 1 Homomorphisms De nition 1 A homomorphism between groups 0 GI with operation ltgt and 0 G2 with operation 0 is a tnction ffrom 01 to G2 such that VI y 6 G1 fzoy fz 0 y Notice that fz ltgt y need not always be equal to fz o For example suppose GI G2 R the set of real numbers under addition That is ltgt and 0 both correspond to addition Suppose f R A R fx 12 Then fz ltgt y z y2 and fz o fy x2 y2 In general fz ltgt y y fz o fy and f is not a homomorphism The following are example of homomorphisms Example 1 Let GI G2 R the set ofreal numbers under addition That is ltgt and 0 both correspond to addition Let f R A R fx lezfor any h E R Then f is a homomorphism because fIltgty kry lezley frfy f10fy ExampleZ Let Gl G2 R 0 the set ofnonzero real numbers under multiplication That is ltgt and 0 both correspond to multiplication Let f R A R fx z for any n E Z48 the positive integers Then f is a homomorphism because frltgty Way Iniyn f1 fy f10 y CSCI 124 and CSCI 224VoIaGWU 2 Example 3 Let Gl G2 Zm the set ofintegers modulo m Letltgt and 0 both correspond to addition mod m Let f 01 A G2 fx 1m mod mfor any h E Zm Then f is a homomorphism because fIltgty krym0dm hr mod m ky mod m fr y f10 y Example4 Let Gl G2 Z the set of nonzero integers modulo p where p is prime Recall that this is a group under multiplication modulo p Let ltgt and 0 both correspond to multiplication modulo p Let f G1 A G2 fx 1 mod pfor any n E Z the positive integers Then f is a homomorphism because fr ltgt y i 9 mad 10 1 mad PMy mod 0 f IAfy f I 0 f y In the examples we ve seen so far ltgt and o are the same operation We now see an example where ltgt and o are different operations Examples Let 01 be the group of integers Z with ltgt being the addition operation Let G2 be the in nite group Hi2 2 p l 17 p1 paw pnin E Z with o the multiplication operation f 01 A G2 fn p is a homomorphism because frltgty if Pit3y frify frofy 2 Properties of Homomorphisms The following are properties of homomorphism f 1 fe E where e is the identity in G1 and E that in G2 This is because fI 06 fI0f Hence f e is the identity in G2 2 fz 1 fz 1 This is to be shown for the HW CSCI 124 and CSCI 224VoIaGWU 3 3 Isomorphisms An isomorphism is a homomorphism that is oneto one For example the following are isomorphisms Example 6 Let GI G2 R the set ofreal numbers under addition That is ltgt and 0 both correspond to addition Let f R A R fx he for any h E R Then f is a isomorphism because rst it is a homomorphism as shown in example 1 above Second it is onetoone This can be proved by contradiction Suppose f is not onetoone Then there are two values x1 and 12 x1 y 12 such that f11 fzg That is ll 12 kzl kzg kilkzl killer 11 12 which provides a contradiction Hence f is onetoone and an isomorphism Example 7 The identity is an isomorphism Consider any group G with any operation ltgt LetGl G2 G Let f be the identity on G That is f G1 A G2 fx I Then f is an isomorphism because rst it is a homomorphism may my fltzgtltgtfltygt fltzgtofltygt Further it is onetoone This can be proved by contradiction Suppose f is not onetoone Then there are two values x1 and 12 11 y 12 such that f11 fltI2 That is 11 12 11 12 which provides a contradiction Hence f is onetoone and an isomorphism

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