# Maths - Size (Order) of Groups

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 Cn 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 e m
e
e m
m m e

This is often called C2, the cyclic group of order two. Equivilantly it is also Z2, the group of integers modulo two.

## Order 3

Cayley Graph Table e r
e
e r
r r e
e r

There is also only one group of order 3 this is C3.

## Order 4

Cayley Graph Table Table (isomorphism) e r
e
e r
r r e
e r
e r
e r
e
e r
e r
r r e
r e e v h vh
e
e v h vh
v v e vh h
h h vh e v
vh vh h v e
e r
e
e r
e r
r r e
r e

There are two possible groups of order 4:

• C4 = C2 C2
• V4 = C2×C2

## Order 6

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.      Fearless Symmetry - This book approaches symmetry from the point of view of number theory. It may not be for you if you are only interested in the geometrical aspects of symmetry such as rotation groups but if you are interested in subjects like modulo n numbers, Galois theory, Fermats last theorem, to name a few topics the chances are you will find this book interesting. It is written in a friendly style for a general audience but I did not find it dumbed down. I found a lot of new concepts to learn. It certainly gives a flavor of the complexity of the subject and some areas where maths is still being discovered.