Groups - definition and basic properties

Next we give two examples of finite groups. For a finite group
we denote by the number of elements in .
A finite group can be given by its *multiplication table*
(also called the *Cayley table*).
This is a square table of size ; the rows and columns
are indexed by the elements of ; the entry in the row
and column is .

We are now going to list some basic consequences of our defining axioms for groups.
First of all, we note that associativity implies that
in a product of any number of elements
in that order, the arrangement of brackets does not matter.
For example, we have
, since

We can therefore omit the brackets altogether and write simply . (A rigorous proof of this is a somewhat tedious induction on .) This, in turn means that we can use the power notation:

The existence of inverses implies that we can extend this (as we do in ) to

With this in mind, we have the following natural rules:

The proof follows straight from the definition, but one has to consider all possible cases, depending on the signs of and .

Edmund F Robertson

11 September 2006