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

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Presentation transcript:

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

Are Z 5 and S = {0,2,4,6,8}  Z 10 the same? · ·

Using colors to decide… · ·

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!

Start with empty tables and fill in based on color… · ·

Since 6+6=2 in S, 2 is yellow… · ·

It follows that 8 is blue and 4 is purple · ·

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!