Group Theory - Axioms
Not every set with an associated operation on its elements is a group. To form a group the set and operation (or rule of combination) must satisfy the following four axioms:
The combination of any two elements (according to the rule of combination of the set) must itself be an element in the set.
Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is, rearranging the parentheses in such an expression will not change its value. Consider, for instance, the following equations:
Consider the first equation. Even though the parentheses were rearranged (the left side requires adding 5 and 2 first, then adding 1 to the result, whereas the right side requires adding 2 and 1 first, then 5), the value of the expression was not altered. Since this holds true when performing addition on any real numbers, we say that “addition of real numbers is an associative operation.”
Associativity is not to be confused with commutativity. Commutativity justifies changing the order or sequence of the operands within an expression while associativity does not.
The set must contain an element which, when combined (according to the operation of the set) with any other element, results in that second element. For example, for the set of whole numbers under the operation of multiplication the identity element is the number 1 since any number multiplied by 1 returns the same number.
4 - Invertibility
For each element in the set there must be one and only one element also in the set which, when combined (according to the operation of the set), results in the identity element.
In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can ‘undo’ the effect of combination with another given element. While the precise definition of an inverse element varies depending on the algebraic structure involved, these definitions coincide in a group.