Abstract Algebra/Group tables

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The Group of Order 2[edit]

Here is the group table for the only group of order 2

Z2
+ 0 1
0 0 1
1 1 0

The Group of Order 3[edit]

Here is the group table for the only group of order 3

Z3
+ 0 1 2
0 0 1 2
1 1 2 0
2 2 0 1

The Groups of Order 4[edit]

Here are the group tables for the only groups of order 4

The cyclic group of order 4[edit]

Two ways of documenting the same group structure
Z4
+ 0 1 2 3
0 0 1 2 3
1 1 2 3 0
2 2 3 0 1
3 3 0 1 2
× 1 2 3 4
1 1 2 3 4
2 2 4 1 3
3 3 1 4 2
4 4 3 2 1
To see more clearly that these two tables actually have the same
group structure you'll need to rename the entries
      0       maps to       1      
1 maps to 2
2 maps to 4
3 maps to 3
1 + 2 = 3 maps to 2 × 4 = 3

Notice that regardless of the way we notate this group, there is an element that generates the whole group.



The other group of order 4[edit]

For the following example, image the number 0 through 3 written in binary, then add the digits without any carrying. For example,

 2 +  3 
10 + 11
   01 
   1

Since binary addition (without carry) is isomorphic to we view this group as being two copies of joined together. That's where the name comes from.

+ 0 1 2 3
0 0 1 2 3
1 1 0 3 2
2 2 3 0 1
3 3 2 1 0

The Group of Order 5[edit]

+ 0 1 2 3 4
0 0 1 2 3 4
1 1 2 3 4 0
2 2 3 4 0 1
3 3 4 0 1 2
4 4 0 1 2 3

Other small groups[edit]

A list of groups of order 1 through 31 compiled by John Pedersen, Dept of Mathematics, University of South Florida [1]

A list of groups names and some examples of group graphs from Wolfram, makers of Mathematica. [2]