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

maps to

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,

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]