# 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
${\displaystyle Z_{5}^{*}}$
× 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.

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


### 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 ${\displaystyle Z_{2}}$ we view this group as being two copies of ${\displaystyle Z_{2}}$ joined together. That's where the name comes from.

${\displaystyle Z_{2}\oplus Z_{2}}$
+ 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

${\displaystyle Z_{5}}$
+ 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]