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
Notice that regardless of the way we notate this group, there is an element that generates the whole group. $$1 + 1 \to 2 \quad 1 + 2 \to 3 \quad 1 + 3 \to 0$$ $$ 2{}\times{} 2 \to 4 \quad 2 \times 4 \to 3 \quad 2 \times 3 \to 1 $$

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 $$Z_2$$ we view this group as being two copies of $$Z_2$$ joined together. That's where the name comes from.

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.