Isomorphism: A First Example MAT 320 Spring 2008 Dr. Hamblin
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Isomorphism: A First Example
MAT 320 Spring 2008Dr. Hamblin
Are Z5 and S = {0,2,4,6,8} Z10 the same?
+ 0 1 2 3 4 · 0 1 2 3 4
0 0 1 2 3 4 0 0 0 0 0 0
1 1 2 3 4 0 1 0 1 2 3 4
2 2 3 4 0 1 2 0 2 4 1 3
3 3 4 0 1 2 3 0 3 1 4 2
4 4 0 1 2 3 4 0 4 3 2 1
+ 0 2 4 6 8 · 0 2 4 6 8
0 0 2 4 6 8 0 0 0 0 0 0
2 2 4 6 8 0 2 0 4 8 2 6
4 4 6 8 0 2 4 0 8 6 4 2
6 6 8 0 2 4 6 0 2 4 6 8
8 8 0 2 4 6 8 0 6 2 8 4
Using colors to decide…
+ 0 1 2 3 4 · 0 1 2 3 4
0 0 1 2 3 4 0 0 0 0 0 0
1 1 2 3 4 0 1 0 1 2 3 4
2 2 3 4 0 1 2 0 2 4 1 3
3 3 4 0 1 2 3 0 3 1 4 2
4 4 0 1 2 3 4 0 4 3 2 1
+ 0 2 4 6 8 · 0 2 4 6 8
0 0 2 4 6 8 0 0 0 0 0 0
2 2 4 6 8 0 2 0 4 8 2 6
4 4 6 8 0 2 4 0 8 6 4 2
6 6 8 0 2 4 6 0 2 4 6 8
8 8 0 2 4 6 8 0 6 2 8 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 1S = 6• Since 1 in Z5 was colored green, this means our
coloring was wrong!
Start with empty tables and fill in based on color…
+ 0 1 2 3 4 · 0 1 2 3 4
0 0 1 2 3 4 0 0 0 0 0 0
1 1 2 3 4 0 1 0 1 2 3 4
2 2 3 4 0 1 2 0 2 4 1 3
3 3 4 0 1 2 3 0 3 1 4 2
4 4 0 1 2 3 4 0 4 3 2 1
+ 0 6 · 0 6
0 0 6 0 0 0
6 6 6 0 6
6 6
6 6
6 6
Since 6+6=2 in S, 2 is yellow…
+ 0 1 2 3 4 · 0 1 2 3 4
0 0 1 2 3 4 0 0 0 0 0 0
1 1 2 3 4 0 1 0 1 2 3 4
2 2 3 4 0 1 2 0 2 4 1 3
3 3 4 0 1 2 3 0 3 1 4 2
4 4 0 1 2 3 4 0 4 3 2 1
+ 0 6 2 · 0 6 2
0 0 6 2 0 0 0 0
6 6 2 6 0 6 2
2 2 6 2 0 2 6
6 6 2
6 2 6
It follows that 8 is blue and 4 is purple
+ 0 1 2 3 4 · 0 1 2 3 4
0 0 1 2 3 4 0 0 0 0 0 0
1 1 2 3 4 0 1 0 1 2 3 4
2 2 3 4 0 1 2 0 2 4 1 3
3 3 4 0 1 2 3 0 3 1 4 2
4 4 0 1 2 3 4 0 4 3 2 1
+ 0 6 2 8 4 · 0 6 2 8 4
0 0 6 2 8 4 0 0 0 0 0 0
6 6 2 8 4 0 6 0 6 2 8 4
2 2 8 4 0 6 2 0 2 4 6 8
8 8 4 0 6 2 8 0 8 6 4 2
4 4 0 6 2 8 4 0 4 8 2 6
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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