Abstract Algebra/Group tables
The Group of Order 2
Here is the group table for the only group of order 2
The Group of Order 3
Here is the group table for the only group of order 3
The Groups of Order 4
Here are the group tables for the only groups of order 4
The cyclic group of order 4
|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|
|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
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.
The Group of Order 5
Other small groups
A list of groups of order 1 through 31 compiled by John Pedersen, Dept of Mathematics, University of South Florida 
A list of groups names and some examples of group graphs from Wolfram, makers of Mathematica.