  Xu and Wu (2001) defined the \emph{generalized wordlength pattern} $(A_1,
..., A_k)$ of an arbitrary fractional factorial design (or orthogonal array) on
$k$ factors. They gave a coding-theoretic proof of the property that the design
has strength $t$ if and only if $A_1 = ... = A_t = 0$. The quantities $A_i$ are
defined in terms of characters of cyclic groups, and so one might seek a direct
character-theoretic proof of this result. We give such a proof, in which the
specific group structure (such as cyclicity) plays essentially no role.
Nonabelian groups can be used if the counting function of the design satisfies
one assumption, as illustrated by a couple of examples.
