Heights of algebraic numbers modulo multiplicative group actions

Given a number field K and a subgroup G⊂K∗ of the multiplicative group of K, Silverman defined the G-height H(θ;G) of an algebraic number θ asH(θ;G):=infg∈G,n∈N{H(g1/nθ)}, where H on the right is the usual absolute height. When G=EK is the units of K, such a height was introduced by Bergé and Martinet who found a formula for H(θ;EK) involving a curious product over the archimedean places of K(θ). We take the analogous product over all places of K(θ) and find that it corresponds to H(θ;K1), where K1 is the kernel of the norm map from K∗ to Q∗. We also find that a natural modification of this same product leads to H(θ;K∗). This is a height function on algebraic numbers which is unchanged under multiplication by K∗. For G=K1, or G=K∗, we show that H(θ;G)=1 if and only if θn∈G for some positive integer n. For these same G we also show that G-heights have the expected finiteness property: for any real number X and any integer N there are, up to multiplication by elements of G, only finitely many algebraic numbers θ such that H(θ;G)