Pdivisible

pdivisible, written with or without a hyphen as p-divisible, is a term used in mathematics with two related meanings that appear in abelian group theory and algebraic geometry.

In abelian group theory, a group G (written additively) is p-divisible if the multiplication-by-p map [p]: G -> G is surjective; equivalently, for every g in G there exists x with p x = g. This means every element has a p-th root. The property for a fixed prime p is weaker than full divisibility, which requires surjectivity for all positive integers n.

Examples and properties: The additive group of rational numbers Q is p-divisible for every prime p. The Prüfer p-group Z(p^∞) is p-divisible and serves as a canonical example of a p-divisible group with torsion. The direct sum (and product) of p-divisible groups is p-divisible, but subgroups need not be; a subgroup can fail to be p-divisible, as multiplication by p may not be surjective there.

In algebraic geometry, a p-divisible group (also called a Barsotti–Tate group) is an inductive system {G_n} of finite commutative group schemes over a base scheme S, with each G_n of order p^{n h} for some height h, and with compatible transition maps that intertwine with multiplication by p. The p-power-torsion of an abelian variety gives a natural example. P-divisible groups play a central role in Dieudonné theory and p-adic Hodge theory, and they are classified up to isogeny by their height and dimension.

See also: divisible group, Barsotti–Tate group, Dieudonné theory, p-adic Hodge theory.