Divisible groups

Abstract

A group G is called divisible if every equation of the form nx = g, with n ∈ N* and g ∈ G, has a solution in G. If this holds only for those n which are powers of a fixed prime p, we obtain the notion of p-divisible group. A group R is reduced if it has no nonzero divisible subgroups. Note that if the equation nx = g has solutions then we also say that n divides g, and we write n|g. Obviously, if n and ord(g) are coprime then n|g. Clearly if g ∈ G, p ∈ P, and k ∈ N then p ^k|g if and only if k ≥ h _p (g).