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Abstract This study demonstrates formal connections between Abelian group theory and factorial experimental designs. I begin by showing that full-factorial designs are isomorphic to elementary p-groups. I continue with an analysis of the general group and subgroup structure of 3 × 3 full-factorial designs commonly utilized in detection studies, followed by extensions to more general but related designs in the subsequent section, namely, 3 × 3 × 3 designs.
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