## Properties of Groups - A Summary

A group is a nonempty set of elements . If then along with a group multiplication operation · (called the product) satisfying the following four conditions

1. Closure. If then is also in 2. Associativity. The group multiplication is associative, a · (b · c) = (a · b) · c

3. The identity, denoted is a member of the group. There exists an element such that for all (The identity is also sometimes denoted I or even simply 1.)

4. Inverses . For every there exists an inverse element such that Note that the term “group multiplication” does not have to denote ordinary multiplication. For example, the set of all integers under addition forms a group where 0 is the identity element and the integer is the ‘inverse’ of the number More or less, the definition of a group is essentially a common sense definition. However, it is remarkable that so much structure can arise out of such a basic definition. This is very typical of mathematics. A definition is used to reduce things down to the least necessary set of fundamental ideas needed to be useful, ie to capture the essence of the system. Then the rest is built up on top of the definition through theorems and such.

The above four properties can be represented compactly in a group table, shown below for the group G consisting of the elements ·                                         a^2 r      Several points must be noted:

Each element appears once in each row and each column, and none but the six elements of the group appear in the table.

Because the identity element appears once in each column, every element has an inverse.

The result of multiplying any element with the identity is the element itself. 