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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.
Somewhat akin to the dimension of vector spaces, every abelian group has a rank. It is defined as the maximal cardinality of a set of linearly independent (over the integers) elements of the group. [10]: 49–50 Finite abelian groups and
If G is a finite non-abelian simple group (e.g. G = A n, the alternating group, for n > 4) then rank(G) = 2. This fact is a consequence of the Classification of finite simple groups . If G is a finitely generated group and Φ( G ) ≤ G is the Frattini subgroup of G (which is always normal in G so that the quotient group G /Φ( G ) is defined ...
The rank of an abelian group is defined as the rank of a free abelian subgroup of for which the quotient group / is a torsion group. Equivalently, it is the cardinality of a maximal subset of G {\displaystyle G} that generates a free subgroup.
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 .
Stated differently the fundamental theorem says that a finitely generated abelian group is the direct sum of a free abelian group of finite rank and a finite abelian group, each of those being unique up to isomorphism. The finite abelian group is just the torsion subgroup of G.
Every elementary abelian p-group is a vector space over the prime field with p elements, and conversely every such vector space is an elementary abelian group. By the classification of finitely generated abelian groups, or by the fact that every vector space has a basis, every finite elementary abelian group must be of the form (Z/pZ) n for n a ...
An object in Ab is injective if and only if it is a divisible group; it is projective if and only if it is a free abelian group. The category has a projective generator (Z) and an injective cogenerator (Q/Z). Given two abelian groups A and B, their tensor product A⊗B is defined; it is again an abelian group.