Generator-offset property: Difference between revisions
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Σ2: [some mos] | Σ2: [some mos] | ||
Σ3: [some mos] | Σ3: [some mos] | ||
(Note: We say “assume” as shorthand for either “assume without loss of generality” or “assume after excluding other possibilities”.) | |||
Assume that S2's generator is also a k-step, and that Σ2's I is located at index n. Then Σ1 and Σ2 are the same mos and even the same mode. Assume the L of S1 (it could be s, but it doesn’t matter) is the result of identifying a and b, and all instances of s in S1 come from c. Then the corresponding steps of S2 must be either all a’s or all b~c’s. Thus these steps are all a’s or all b’s. This contradicts the assumption that S has exactly three step sizes. This shows that the maximum variety of S must be at least 3, and that P has at least two preimages in S. | Assume that S2's generator is also a k-step, and that Σ2's I is located at index n. Then Σ1 and Σ2 are the same mos and even the same mode. Assume the L of S1 (it could be s, but it doesn’t matter) is the result of identifying a and b, and all instances of s in S1 come from c. Then the corresponding steps of S2 must be either all a’s or all b~c’s. Thus these steps are all a’s or all b’s. This contradicts the assumption that S has exactly three step sizes. This shows that the maximum variety of S must be at least 3, and that P has at least two preimages in S. | ||