Properties of Groups - A Summary
A groupis a nonempty set of elements . Ifthen along with a group multiplication operation · (called the product) satisfying the following four conditions
1. Closure. Ifthenis also in
2. Associativity. The group multiplication is associative, a · (b · c) = (a · b) · c
3. The identity, denotedis a member of the group. There exists an elementsuch thatfor all(The identity is also sometimes denoted I or even simply 1.)
4. Inverses . For everythere exists an inverse elementsuch 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 integeris 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
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 elementappears once in each column, every element has an inverse.
The result of multiplying any element with the identity is the element itself.