Abstract Algebra/Group tables
The Group of Order 2[edit]
Here is the group table for the only group of order 2
+  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
+  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  

 
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]