  We present a criterion for proving that certain groups of the form $\mathbb
Z/m\mathbb Z\oplus\mathbb Z/n\mathbb Z$ do not occur as the torsion subgroup of
any elliptic curve over suitable (families of) number fields. We apply this
criterion to eliminate certain groups as torsion groups of elliptic curves over
cubic and quartic fields. We also use this criterion to give the list of all
torsion groups of elliptic curves occurring over a specific cubic field and
over a specific quartic field.
