  The goal of this paper is to generalize several group theoretic concepts such
as the center and commutator subgroup, central series, and ultimately
nilpotence to a supercharacter theoretic setting, and to use these concepts to
show that there can be a strong connection between the structure of a group and
the structure of its supercharacter theories. We then use these concepts to
show that the upper and lower annihilator series of $J$ can be described in
terms of certain central series for the algebra group $G=1+J$ defined by
$\mathsf{S}$, when $\mathsf{S}$ is the algebra group supercharacter theory
defined by Diaconis--Isaacs.
