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A group is a cancellable complete flat combiner. This combination of properties happens to arise naturally in a wide variety of contexts; and it determines enough about the binary operator (and the values on which it acts) to make for a rich set of properties that groups may posess and rich structure in the hormomorphisms between groups. Any non-empty group necessarily has an identity value (that the binary operator combines with any other to give that other) and, for each value, an inverse (that, when combined with it, gives the identity); indeed, orthodoxy defines groups by these properties (thereby excluding the empty group; this is all of the practical difference between the two definitions).
In any category of homomorphisms of some mathematical structure-type T, one can single out the category of T-isomorphisms; these include T-autoisomorphisms, which in turn include the identity morphisms. The T-autoisomorphisms of any identity form a group under composition (with the identity T-autoisomorphism serving as the identity operand of composition) and each T-isomorphism q, with inverse b, induces a group-isomorphism between the groups of T-autoisomorphisms of the identities b&on;q and q&on;b at each end, namely (: b&on;f&on;q ←f :) with inverse (: q&on;h&on;b ←h :) – we can map b to one of these and q to the other, with free choice as to which way round (neither is more natural; and each is equivalent). When the T-isomorphism, q here, is a T-autoisomorphism of some identity, let G be the group of T-autoisomorphisms of that identity and let E be the group of group-autoisomorphisms (G: |G); then the construction just given has mapped q and its inverse (both members of G) to a mutually inverse pair of E's members; we thus get a mapping (E: |G) that lets us represent T-autoisomorphisms as group-automorphisms; in this sense, groups are universal for autoisomorphisms – any category's autoisomorphisms can be represented as a functor's image in a category of group isomorphisms. As a result, the study of groups (and their homomorphisms) can tell us a great deal about other categories generally.
This page is woefully incomplete; over time, I hope I shall find time to augment it with many more; even so, I shall but scratch the surface of this vast and important subject.
The theory of groups was developed – independently and roughly simultaneously, in the early 19th century – by Niels Henrik Abel (1802–1829) and Évariste Galois (1811–1832), both of whom died tragically young, their work unappreciated until after their deaths; by the time folk began to recognise the enormous worth of what both had done, it was too late to give either the recognition each deserved.
The advent of group theory began so great a revolution on how folk thought about mathematics that one may fairly call it a rebirth and a new beginning. At the same time, the particular form in which it was first presented (in terms of identity and inverses) lead to its use in some contexts (I particularly think of ring theory) where only cancellation (and an understanding of what must be taken into account when incomplete) is really needed for the development of the essence of a subject (e.g. the fundamental theorem of arithmetic); which has lead to some encumbrance (with analysis burdened by case-splitting on sign, for example) that I hope I may be able to avoid by my re-casting of the axioms of this field – which, further, I hope may make the exposition of group theory itself clearer, by addressing more directly the two facts about a group that actually do most of the work that attends proofs of its results.
Written by Eddy.