height (abelian group)
   HOME

TheInfoList



OR:

In mathematics, the height of an element ''g'' of an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
''A'' is an invariant that captures its divisibility properties: it is the largest
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
''N'' such that the equation ''Nx'' = ''g'' has a solution ''x'' ∈ ''A'', or the symbol ∞ if there is no such ''N''. The ''p''-height considers only divisibility properties by the powers of a fixed
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
''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 In mathematics, two Prüfer theorems, named after Heinz Prüfer, describe the structure of certain infinite abelian groups. They have been generalized by L. Ya. Kulikov. Statement Let ''A'' be an abelian group. If ''A'' is finitely generated ...
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 Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for ...
: * ''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 In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists a ...
. 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 mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
in ''A''; in particular, subquotients of the filtration are isomorphism invariants of ''A''.


Ulm subgroups

Let ''p'' be a fixed prime number. The (first) Ulm subgroup of an abelian group ''A'', denoted ''U''(''A'') or ''A''1, is ''p''''ω''''A'' = ∩''n'' ''p''''n''''A'', where ''ω'' is the smallest infinite ordinal. It consists of all elements of ''A'' of infinite height. The family of Ulm subgroups indexed by ordinals ''σ'' is defined by transfinite induction: * ''U''0(''A'') = ''A''; * ''U''''σ''+1(''A'') = ''U''(''U''''σ''(''A'')); * ''U''''τ''(''A'') = ∩''σ'' < ''τ'' ''U''''σ''(''A'') if ''τ'' is a
limit ordinal In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists a ...
. Equivalently, ''U''''σ''(''A'') = ''p''''ωσ''''A'', where ''ωσ'' is the product of ordinals ''ω'' and ''σ''. Ulm subgroups form a decreasing filtration of ''A'' whose quotients ''U''''σ''(''A'') = ''U''''σ''(''A'')/''U''''σ''+1(''A'') are called the Ulm factors of ''A''. This filtration stabilizes and the smallest ordinal ''τ'' such that ''U''''τ''(''A'') = ''U''''τ''+1(''A'') is the Ulm length of ''A''. The smallest Ulm subgroup ''U''''τ''(''A''), also denoted ''U''(''A'') and ''p''A, consists of all ''p''-divisible elements of ''A'', and being
divisible group In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an ''n''th multiple for each positive in ...
, it is a direct summand of ''A''. For every Ulm factor ''U''''σ''(''A'') the ''p''-heights of its elements are finite and they are unbounded for every Ulm factor except possibly the last one, namely ''U''''τ''−1(''A'') when the Ulm length ''τ'' is a
successor ordinal In set theory, the successor of an ordinal number ''α'' is the smallest ordinal number greater than ''α''. An ordinal number that is a successor is called a successor ordinal. Properties Every ordinal other than 0 is either a successor ordin ...
.


Ulm's theorem

The second Prüfer theorem provides a straightforward extension of the
fundamental theorem of finitely generated abelian groups In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...
to countable abelian ''p''-groups without elements of infinite height: each such group is isomorphic to a direct sum of cyclic groups whose orders are powers of ''p''. Moreover, the cardinality of the set of summands of order ''p''''n'' is uniquely determined by the group and each sequence of at most countable cardinalities is realized.
Helmut Ulm Helmut Ulm (born 21 June 1908 in Gelsenkirchen; died 13 June 1975) was a German mathematician who established the classification of countable periodic abelian groups by means of their Ulm invariants. Career Helmut Ulm's father was an elementary ...
(1933) found an extension of this classification theory to general countable ''p''-groups: their isomorphism class is determined by the isomorphism classes of the Ulm factors and the ''p''-divisible part. : Ulm's theorem. ''Let'' ''A'' ''and'' ''B'' ''be countable abelian'' ''p''-''groups such that for every ordinal'' ''σ'' ''their Ulm factors are isomorphic'', ''U''''σ''(''A'') ≅ ''U''''σ''(''B'') ''and the'' ''p''-''divisible parts of'' ''A'' ''and'' ''B'' ''are isomorphic'', ''U''(''A'') ≅ ''U''(''B''). ''Then'' ''A'' ''and'' ''B'' ''are isomorphic.'' There is a complement to this theorem, first stated by Leo Zippin (1935) and proved in Kurosh (1960), which addresses the existence of an abelian ''p''-group with given Ulm factors. : ''Let'' ''τ'' ''be an ordinal and'' ''be a family of countable abelian'' ''p''-''groups indexed by the ordinals'' ''σ'' < ''τ'' ''such that the'' ''p''-''heights of elements of each'' ''A''''σ'' ''are finite and, except possibly for the last one, are unbounded. Then there exists a reduced abelian'' ''p''-''group'' ''A'' ''of Ulm length'' ''τ'' ''whose Ulm factors are isomorphic to these'' ''p''-''groups'', ''U''''σ''(''A'') ≅ ''A''''σ''. Ulm's original proof was based on an extension of the theory of
elementary divisors In algebra, the elementary divisors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain. If R is a PID and M a finitely generated R-module, then '' ...
to infinite
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
.


Alternative formulation

George Mackey George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American mathematician known for his contributions to quantum logic, representation theory, and noncommutative geometry. Career Mackey earned his bachelor of arts at Rice Unive ...
and
Irving Kaplansky Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and amateur musician.O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St Andr ...
generalized Ulm's theorem to certain
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
over a
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
discrete valuation ring In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions: # ''R'' i ...
. They introduced invariants of abelian groups that lead to a direct statement of the classification of countable periodic abelian groups: given an abelian group ''A'', a prime ''p'', and an ordinal ''α'', the corresponding ''α''th Ulm invariant is the dimension of the quotient : ''p''''α''''A'' 'p''''p''''α''+1''A'' 'p'' where ''B'' 'p''denotes the ''p''-torsion of an abelian group ''B'', i.e. the subgroup of elements of order ''p'', viewed as a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over the
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
with ''p'' elements. : ''A countable periodic reduced abelian group is determined uniquely up to isomorphism by its Ulm invariants for all prime numbers ''p'' and countable ordinals ''α''.'' Their simplified proof of Ulm's theorem served as a model for many further generalizations to other classes of abelian groups and modules.


References

* László Fuchs (1970), ''Infinite abelian groups, Vol. I''. Pure and Applied Mathematics, Vol. 36. New York–London: Academic Press *
Irving Kaplansky Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and amateur musician.O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St Andr ...
and
George Mackey George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American mathematician known for his contributions to quantum logic, representation theory, and noncommutative geometry. Career Mackey earned his bachelor of arts at Rice Unive ...
, ''A generalization of Ulm's theorem''. Summa Brasil. Math. 2, (1951), 195–202 * * {{cite journal , last1 = Ulm , first1 = H , year = 1933 , title = Zur Theorie der abzählbar-unendlichen Abelschen Gruppen , journal = Math. Ann. , volume = 107 , pages = 774–803 , JFM=59.0143.03 , doi=10.1007/bf01448919 Abelian group theory Infinite group theory