On a Galois property of fields generated by the torsion of an abelian variety

In this article, we study a certain Galois property of subextensions of $k(A_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}})$, the minimal field of definition of all torsion points of an abelian variety $A$ defined over a number field $k$. Concretely, we show that each subfield of $k(A_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}})$ which is Galois over $k$ (of possibly infinite degree) and whose Galois group has finite exponent is contained in an abelian extension of some finite extension of $k$. As an immediate corollary of this result and a theorem of Bombieri and Zannier, we deduce that each such field has the Northcott property, i.e. does not contain any infinite set of algebraic numbers of bounded height.