  In this article, we study the minimal degree [K(T):K] of a p-subgroup T <=
E(\overline{K})_tors for an elliptic curve E/K defined over a number field K.
Our results depend on the shape of the image of the p-adic Galois
representation \rho_{E,p^infty}:Gal_K-->GL(2,Z_p). However, we are able to show
that there are certain uniform bounds for the minimal degree of definition of
T. When the results are applied to K=Q and p=2, we obtain a divisibility
condition on the minimal degree of definition of any subgroup of E[2^n] that is
best possible.
