Here we will categorise how many and what type of groups there are of a given size (known as 'order' in the terminology. There is at least one group of a given size, that is the cyclic group, for instance if we are looking for a group of order 'n' then C_{n }is such a group. There may, or may not, be other groups of order n.
Order 1
Every group must contain the identity element so since the group only contains one element this must be the identity element.
Order 2
There is only one valid group of order 2, all other groups of order 2 are isomorpic to this.
Cayley Graph  Table  

This is often called C_{2}, the cyclic group of order two. Equivilantly it is also Z_{2}, the group of integers modulo two.
Order 3
Cayley Graph  Table  

There is also only one group of order 3 this is C_{3}.
Order 4
Cayley Graph  Table  Table (isomorphism)  





There are two possible groups of order 4:
 C_{4} = C_{2}C_{2}
 V_{4} = C_{2}×C_{2}
Order 5
Order 6