Abstract Algebra/Group tables

The Group of Order 2

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

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

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
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				$Z_{5}^{*}$ × 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.

$1+1\to 2\quad 1+2\to 3\quad 1+3\to 0$
$2{}\times {}2\to 4\quad 2\times 4\to 3\quad 2\times 3\to 1$

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.

$Z_{2}\oplus Z_{2}$

+	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

$Z_{5}$

+	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

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]