Editing Group theory (section)

===Abstract groups===
Most groups considered in the first stage of the development of group theory were "concrete", having been realized through numbers, permutations, or matrices. It was not until the late nineteenth century that the idea of an '''abstract group''' began to take hold, where "abstract" means that the nature of the elements are ignored in such a way that two [[group isomorphism|isomorphic groups]] are considered as the same group. A typical way of specifying an abstract group is through a [[presentation of a group|presentation]] by ''generators and relations'',

: <math> G = \langle S|R\rangle. </math>

A significant source of abstract groups is given by the construction of a ''factor group'', or [[quotient group]], ''G''/''H'', of a group ''G'' by a [[normal subgroup]] ''H''. [[Class group]]s of [[algebraic number field]]s were among the earliest examples of factor groups, of much interest in [[number theory]]. If a group ''G'' is a permutation group on a set ''X'', the factor group ''G''/''H'' is no longer acting on ''X''; but the idea of an abstract group permits one not to worry about this discrepancy.

The change of perspective from concrete to abstract groups makes it natural to consider properties of groups that are independent of a particular realization, or in modern language, invariant under [[isomorphism]], as well as the classes of group with a given such property: [[finite group]]s, [[periodic group]]s, [[simple group]]s, [[solvable group]]s, and so on. Rather than exploring properties of an individual group, one seeks to establish results that apply to a whole class of groups. The new paradigm was of paramount importance for the development of mathematics: it foreshadowed the creation of [[abstract algebra]] in the works of [[David Hilbert|Hilbert]], [[Emil Artin]], [[Emmy Noether]], and mathematicians of their school.{{citation needed|date=June 2012}}