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. 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: 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. 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.