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In mathematics, the height of an element ''g'' of an abelian group ''A'' is an invariant that captures its divisibility properties: it is the largest natural number ''N'' such that the equation ''Nx'' = ''g'' has a solution ''x'' ∈ ''A'', or symbol ∞ if the largest number with this property does not exist. The ''p''-height considers only divisibility properties by the powers of a fixed prime number ''p''. The notion of height admits a refinement so that the ''p''-height becomes an ordinal number. Height plays an important role in Prüfer theorems and also in Ulm's theorem, which describes the classification of certain infinite abelian groups in terms of their Ulm factors or Ulm invariants. == Definition of height == Let ''A'' be an abelian group and ''g'' an element of ''A''. The ''p''-height of ''g'' in ''A'', denoted ''h''''p''(''g''), is the largest natural number ''n'' such that the equation ''p''''n''''x'' = ''g'' has a solution in ''x'' ∈ ''A'', or the symbol ∞ if a solution exists for all ''n''. Thus ''h''''p''(''g'') = ''n'' if and only if ''g'' ∈ ''p''''n''''A'' and ''g'' ∉ ''p''''n''+1''A''. This allows one to refine the notion of height. For any ordinal ''α'', there is a subgroup ''p''''α''''A'' of ''A'' which is the image of the multiplication map by ''p'' iterated ''α'' times, defined using transfinite induction: * ''p''0''A'' = ''A''; * ''p''''α''+1''A'' = ''p''(''p''''α''''A''); * ''p''''β''''A''=∩''α'' < ''β'' ''p''''α''''A'' if ''β'' is a limit ordinal. The subgroups ''p''''α''''A'' form a decreasing filtration of the group ''A'', and their intersection is the subgroup of the ''p''-divisible elements of ''A'', whose elements are assigned height ∞. The modified ''p''-height ''h''''p''∗(''g'') = ''α'' if ''g'' ∈ ''p''''α''''A'', but ''g'' ∉ ''p''''α''+1''A''. The construction of ''p''''α''''A'' is functorial in ''A''; in particular, subquotients of the filtration are isomorphism invariants of ''A''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Height (abelian group)」の詳細全文を読む スポンサード リンク
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