Defining contrast subgroup ($\mathcal{H}=⟨G_1,\dots ,G_k⟩$)

Factor subsets multiply by symmetric difference; their sign columns multiply pointwise. The subgroup generated by independent defining words has $2^k$ elements. On a selected fractional factorial design, every word in this subgroup is constant, with sign determined by the chosen generator signs. The cosets classify identical or opposite contrast columns.

Two factorial contrasts are aliased if their columns are proportional on the selected fractional factorial design. A word $G$ that has fixed sign $s_G$ gives ${\chi }_{AG}=s_G{\chi }_A$. Conversely, averaging a character over the fraction vanishes unless its word belongs to the defining contrast subgroup, so every alias class is exactly a subgroup coset. In a half fraction each class has two distinct words; their separate coefficients cannot be identified without assumptions or additional runs.