Isomorphism: A First Example

1
Isomorphism A First Example

• MAT 320 Spring 2008
• Dr. Hamblin

2
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
3
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
4
It seems like the answer is no

• Color-coding the elements of each ring shows that
  the multiplication tables dont 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!

5
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
6
Since 662 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
7
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
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!