A History of GroupTheory William Komp History of Mathematics Dr. Davitt
This raises the question of traits do all groups have in common, and as an extension on to this what additional traits do all fields share. There have been many interpretations of the structure of groups. H. Weber's notion of a group has for axioms, but it turns out that one of his axioms is unnecessary when compared to the modern definition. The modern definition of a group is as follows:
Given any set A, we say that A forms a groupunder the binary operation @ if and only if it satisfies the followingfour criteria.
I) Given any two elements in A, then their product under the binary operation @ will be in A [closure property]
II) Given three elements in A, these three elements will satisfy the associative property.
III) In A, there exists an element such thatthe product of this element with any other element in A, namely ©,will generate the element © back again.
IV) There exists an element ©' in A suchthat the product of © and ©' will give the element mentionedin III.(Gallian) Group Theory has enormous potential in practical applications in areas other than that of mathematics. In mathematics, group theory is the basis of real analysis. Namely,the set of R, the real numbers, must be complete and satisfy all of theproperties of an ordered field (Sherwood 14-19). This provides thenecessary algebra to begin proving and developing axiomatically all ofCalculus. The analogy can also be made with the complex numbers,with some increased complexity in the area of complex analysis. The propertiesof a field are very similar to those of a group. | |
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