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# 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 .

Next: Modular arithmetic Up: MT2002 Algebra Previous: Introduction   Contents

Edmund F Robertson

11 September 2006