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- W2078512985 abstract "In what follows, a coset is a subset of a group G of the form aH , where H is a subgroup of G ; H can be recovered from the coset C : it is the only subgroup which is obtained from C by a left translation; we note in passing that these cosets, that we write systematically with the group to the right, are also of the form Ka , since aH = aHa −1 a . A classical combinatorial lemma involving cosets appears in Neumann [1952]: If the coset C = aH is the union of the finite family of cosets C 1 = a 1 H 1 ,…, C n = a n H n , then it is the union of those C i whose corresponding H i has finite index in H . In a structure where a group G is defined, Boolean combinations of cosets modulo its definable subgroups form a family of definable sets (by definable, we mean “definable with parameters”). The situation when any definable set is of that kind has been characterized model-theoretically in Hrushovski and Pillay [1987]: A group G is one-based if and only if, for each n, every definable subset of the cartesian power G n is a(finite!) Boolean combination of cosets modulo definable subgroups . One side is given by a beautiful lemma of Pillay, stating that, in a one-based group which is saturated enough, every type is a right translate of the generic of its left stabilizer." @default.
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- W2078512985 title "The last word on elimination of quantifiers in modules" @default.
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- W2078512985 doi "https://doi.org/10.2307/2274656" @default.
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