= Solution
Code the two levels of each factor as $\pm1$. For a factor subset $A$, its <factorial contrast> column is the character $\chi_A(x)=\prod_{j\in A}x_j$. Products of columns correspond to symmetric differences of subsets, because $x_j^2=1$.
Let $G_1,\ldots,G_k$ be defining words whose incidence vectors are linearly independent over $\mathbb F_2$. Write $x_j=(-1)^{u_j}$, with $u_j\in\mathbb F_2$. Specifying signs $\chi_{G_i}(x)=s_i$ is then a rank-$k$ system of binary linear equations. Every one of its $2^k$ right-hand sides has $2^{n-k}$ solutions. Thus \b[$k$ independent defining contrasts split the full $2^n$ design into $2^k$ equal fractions.] Independence is necessary: dependent defining words do not yield $2^k$ distinct sets.
The <defining contrast subgroup> has $2^k$ words. On a selected fraction its word $G$ has a constant sign $s_G$, so
$$
\chi_{AG}=s_G\chi_A.
$$
Hence the coefficients of $A$ and $AG$ cannot be estimated separately without assumptions or extra runs: this is <aliasing in a fractional factorial design>. Conversely, averaging a character over the affine binary solution space vanishes unless its word is in the defining subgroup, so these signed subgroup cosets give all aliases. Here a lowercase treatment label records the factors at their high levels, and $(1)$ puts all five at their low levels; signs must be retained when listing aliases.
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