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The rank of a symmetry group is closely related to the complexity of the object (a molecule, a crystal structure) being under the action of the group. If G is a crystallographic point group, then rank(G) is up to 3. [9] If G is a wallpaper group, then rank(G) = 2 to 4. The only wallpaper-group type of rank 4 is p2mm. [10]
In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. [1] The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the rational numbers of dimension rank A.
The automorphism group of a free abelian group of finite rank is the general linear group (,), which can be described concretely (for a specific basis of the free automorphism group) as the set of invertible integer matrices under the operation of matrix multiplication.
The free group in two elements is SQ universal; the above follows as any SQ universal group has subgroups of all countable ranks. Any group that acts on a tree, freely and preserving the orientation, is a free group of countable rank (given by 1 plus the Euler characteristic of the quotient graph). The Cayley graph of a free group of finite ...
In biology, taxonomic rank (which some authors prefer to call nomenclatural rank [1] because ranking is part of nomenclature rather than taxonomy proper, according to some definitions of these terms) is the relative or absolute level of a group of organisms (a taxon) in a hierarchy that reflects evolutionary relationships.
Note that a "complex Lie group" is defined as a complex analytic manifold that is also a group whose multiplication and inversion are each given by a holomorphic map. The dimensions in the table below are dimensions over C. Note that every complex Lie group/algebra can also be viewed as a real Lie group/algebra of twice the dimension.
The rank of an abelian group is the dimension of the -vector space .Equivalently it is the maximal cardinality of a linearly independent (over ) subset of .. If is torsion-free then it injects into .
A ranking is a relationship between a set of items, often recorded in a list, such that, ... or league tables group teams of a particular league, conference, ...