  By virtue of conditions (i)-(iv), the transformations of S form a group GS  in the precise sense that this term has in algebra. GS  includes subgroups, i.e., subsets which contain I and satisfy conditions (i) and (iv). If H is a subgroup of GS  and Φ is a feature of S, or of its elements or parts, which is not affected by the transformations of Φ, we say that Φ is H-invariant. The only GS-invariant is the cardinality of S (i.e., the number of elements in the manifold). On the other hand, the group {I}, consisting of the identity alone, trivially preserves every conceivable feature. Between these two extremes there can be many different subgroups with all sorts of interesting invariants, depending on the respective group structure. If S is not an arbitrary (structureless) set, but a numerical manifold as described by Klein, it inherits structure from the real number field, which contributes to characterize the different subgroups of GS  and their invariants. Thus, the group of continuous transformations preserves the topological properties (neighborhood relations), and the group of linear transformations preserves the projective properties. 