# Height (abelian group)

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 the symbol ∞ if there is no such *N*. 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*.

## 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 {*U*^{σ}(*A*)} 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.

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, 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.

## Ulm's theorem

The second Prüfer theorem 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 *p*^{n} 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.

**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*{*A*_{σ}}*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 to infinite matrices.

### Alternative formulation

George Mackey 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

*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 over the finite field 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 MR0255673 - Irving Kaplansky and George Mackey,
*A generalization of Ulm's theorem*. Summa Brasil. Math. 2, (1951), 195–202 MR0049165 - Kurosh, A. G. (1960),
*The theory of groups*, New York: Chelsea, MR 0109842 - Ulm, H (1933). "Zur Theorie der abzählbar-unendlichen Abelschen Gruppen".
*Math. Ann*.**107**: 774–803. doi:10.1007/bf01448919. JFM 59.0143.03.