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Isomorphism: A First Example MAT 320 Spring 2008 Dr. Hamblin.

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Presentation on theme: "Isomorphism: A First Example MAT 320 Spring 2008 Dr. Hamblin."— Presentation transcript:

1 Isomorphism: A First Example MAT 320 Spring 2008 Dr. Hamblin

2 Are Z 5 and S = {0,2,4,6,8}  Z 10 the same? +01234·01234 001234000000 112340101234 223401202413 334012303142 440123404321 +02468·02468 002468000000 224680204826 446802408642 668024602468 880246806284

3 Using colors to decide… +01234·01234 001234000000 112340101234 223401202413 334012303142 440123404321 +02468·02468 002468000000 224680204826 446802408642 668024602468 880246806284

4 It seems like the answer is no… Color-coding the elements of each ring shows that the multiplication tables don’t match up However, notice something in the multiplication table for S: This shows that 1 S = 6 Since 1 in Z5 was colored green, this means our coloring was wrong!

5 Start with empty tables and fill in based on color… +01234·01234 001234000000 112340101234 223401202413 334012303142 440123404321 +06·06 006000 66606 66 66 66

6 Since 6+6=2 in S, 2 is yellow… +01234·01234 001234000000 112340101234 223401202413 334012303142 440123404321 +062·062 00620000 6626062 2262026 662 626

7 It follows that 8 is blue and 4 is purple +01234·01234 001234000000 112340101234 223401202413 334012303142 440123404321 +06284·06284 006284000000 662840606284 228406202468 884062808642 440628404826

8 With this new coloring… …we see that the two rings have exactly the same structure When two rings have exactly the same addition and multiplication tables (under some correspondence between their elements), we say the rings are isomorphic iso = same, morphic = structure Finding the correspondence is the hard part!

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