Height (abelian group)
Encyclopedia
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.
This allows one to refine the notion of height.
For any ordinal α, there is a subgroup pαA of A which is the image of the multiplication map by p iterated α times, defined using
transfinite induction
:
The subgroups pαA 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 hp∗(g) = α if g ∈ pαA, but g ∉ pα+1A. The construction of pαA is functorial in A; in particular, subquotients of the filtration are isomorphism invariants of A.
Equivalently, Uσ(A) = pωσA, where ωσ is the product of ordinals ω and σ.
Ulm subgroups form a decreasing filtration of A whose quotients Uσ(A) = Uσ(A)/Uσ+1(A) are called the Ulm factors of A. This filtration stabilizes and the smallest ordinal τ such that Uτ(A) = Uτ+1(A) is the Ulm length of A. The smallest Ulm subgroup Uτ(A), also denoted U∞(A) and p∞A, consists of all p-divisible elements of A, and being divisible group
, it is a direct summand of A.
For every Ulm factor Uσ(A) the p-heights of its elements are finite and they are unbounded for every Ulm factor except possibly the last one, namely Uτ−1(A) when the Ulm length τ is a successor ordinal
.
provides a straightforward extension of the fundamental theorem of finitely generated abelian groups 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 pn is uniquely determined by the group and each sequence of at most countable cardinalities is realized. Helmut Ulm
(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.
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.
Ulm's original proof was based on an extension of the theory of elementary divisors
to infinite matrices
.
and Irving Kaplansky
generalized Ulm's theorem to certain modules over a complete discrete valuation ring
. 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 αth Ulm invariant is the dimension of the quotient
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
over the finite field
with p elements.
Their simplified proof of Ulm's theorem served as a model for many further generalizations to other classes of abelian groups and modules.
Abelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...
A is an invariant that captures its divisibility properties: it is the largest natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
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
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
p. The notion of height admits a refinement so that the p-height becomes an ordinal number
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
. Height plays an important role in Prüfer theorems
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...
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 hp(g), is the largest natural number n such that the equation pnx = g has a solution in x ∈ A, or the symbol ∞ if a solution exists for all n. Thus hp(g) = n if and only if g ∈ pnA and g ∉ pn+1A.This allows one to refine the notion of height.
For any ordinal α, there is a subgroup pαA of A which is the image of the multiplication map by p iterated α times, defined using
transfinite induction
Transfinite induction
Transfinite induction is an extension of mathematical induction to well-ordered sets, for instance to sets of ordinal numbers or cardinal numbers.- Transfinite induction :Let P be a property defined for all ordinals α...
:
- p0A = A;
- pα+1A = p(pαA);
- pβA=∩α < β pαA if β is a limit ordinal.
The subgroups pαA 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 hp∗(g) = α if g ∈ pαA, but g ∉ pα+1A. The construction of pαA is functorial 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 A1, is pωA = ∩n pnA, where ω is the smallest infinite ordinal. It consists of all elements of A of infinite height. The family {Uσ(A)} of Ulm subgroups indexed by ordinals σ is defined by transfinite induction:- U0(A) = A;
- Uσ+1(A) = U(Uσ(A));
- Uτ(A) = ∩σ < τ Uσ(A) if τ is a limit ordinal.
Equivalently, Uσ(A) = pωσA, where ωσ is the product of ordinals ω and σ.
Ulm subgroups form a decreasing filtration of A whose quotients Uσ(A) = Uσ(A)/Uσ+1(A) are called the Ulm factors of A. This filtration stabilizes and the smallest ordinal τ such that Uτ(A) = Uτ+1(A) is the Ulm length of A. The smallest Ulm subgroup Uτ(A), also denoted U∞(A) and p∞A, consists of all p-divisible elements of A, and being divisible group
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 nth multiple for each positive integer n...
, it is a direct summand of A.
For every Ulm factor Uσ(A) the p-heights of its elements are finite and they are unbounded for every Ulm factor except possibly the last one, namely Uτ−1(A) when the Ulm length τ is a successor ordinal
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...
.
Ulm's theorem
The second Prüfer theoremPrüfer 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...
provides a straightforward extension of the fundamental theorem of finitely generated abelian groups 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 pn is uniquely determined by the group and each sequence of at most countable cardinalities is realized. Helmut Ulm
Helmut Ulm
Helmut Ulm was a German mathematician who established the classification of countable periodic abelian groups by means of their Ulm invariants.- Short biography :...
(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, Uσ(A) ≅ Uσ(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 {Aσ} be a family of countable abelian p-groups indexed by the ordinals σ < τ such that the p-heights of elements of each Aσ 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, Uσ(A) ≅ Aσ.
Ulm's original proof was based on an extension of the theory of elementary divisors
Elementary divisors
In algebra, the elementary divisors of a module over a principal ideal domain occur in one form of the structure theorem for finitely generated modules over a principal ideal domain....
to infinite matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
.
Alternative formulation
George MackeyGeorge Mackey
George Whitelaw Mackey was an American mathematician. Mackey earned his bachelor of arts at Rice University in 1938 and obtained his Ph.D. at Harvard University in 1942 under the direction of Marshall H. Stone...
and Irving Kaplansky
Irving Kaplansky
Irving Kaplansky was a Canadian mathematician.-Biography:He was born in Toronto, Ontario, Canada, after his parents emigrated from Poland and attended the University of Toronto as an undergraduate. After receiving his Ph.D...
generalized Ulm's theorem to certain modules over a complete discrete valuation ring
Discrete valuation ring
In abstract algebra, a discrete valuation ring is a principal ideal domain 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:...
. 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 αth Ulm invariant is the dimension of the quotient
- pαA[p]/pα+1A[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
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
over the finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
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.