Such a group acts on the "n"-dimensional vector space "K""n" by linear transformations. This action makes matrix groups conceptually similar to permutation groups, and the geometry of the action may be usefully exploited to establish properties of the group "G". Permutation groups and matrix groups are special cases of transformation groups: groups that act on a certain space "X" preserving its inherent structure. In the case of permutation groups, "X" is a set; for matrix groups, "X" is a vector space. The concept of a transformation group is closely related with the concept of a symmetry group: transformation groups frequently consist of "all" transformations that preserve a certain structure. The theory of transformation groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, considers group actions on manifolds by homeomorphisms or diffeomorphisms. The groups themselves may be discrete or continuous. 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 as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by "generators and relations",  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 groups of algebraic number fields 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 groups, periodic groups, simple groups, solvable groups, 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 Hilbert, Emil Artin, Emmy Noether, and mathematicians of their school.