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**1 - 4**of**4**### Second cohomology groups and finite covers

, 909

"... For Ω an infinite set, k ≥ 2 and W the set of k-sets from Ω, there is a natural closed permutation group Γk which is a non-split extension 0 → Z W 2 → Γk → Sym(Ω) → 1. We classify the closed subgroups of Γk which project onto Sym(Ω). The question arises in model theory as a problem about finite cov ..."

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For Ω an infinite set, k ≥ 2 and W the set of k-sets from Ω, there is a natural closed permutation group Γk which is a non-split extension 0 → Z W 2 → Γk → Sym(Ω) → 1. We classify the closed subgroups of Γk which project onto Sym(Ω). The question arises in model theory as a problem about finite covers, but here we formulate and solve it in algebraic terms. 1 Introduction and background on finite covers The problem of understanding the finite covers of a structure arises in model theory (for example, see [1], [9]), but it also has a natural formulation in purely algebraic terms as an extension problem in the category of permutation groups and we adopt this approach here. We begin by reviewing some

### Almost-free finite covers

, 705

"... Let W be a first-order structure and ρ be an Aut(W)-congruence on W. In this paper we define the almost-free finite covers of W with respect to ρ, and we show how to construct them. These are a generalization of free finite covers. A consequence of a result of [5] is that any finite cover of W with ..."

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Let W be a first-order structure and ρ be an Aut(W)-congruence on W. In this paper we define the almost-free finite covers of W with respect to ρ, and we show how to construct them. These are a generalization of free finite covers. A consequence of a result of [5] is that any finite cover of W with binding groups all equal to a simple non-abelian permutation group is almostfree with respect to some ρ on W. Our main result gives a description (up to isomorphism) in terms of the Aut(W)-congruences on W of the kernels of principal finite covers of W with bindings groups equal at any point to a simple non-abelian regular permutation group G. Then we analyze almost-free finite covers of Ω (n) , the set of ordered n-tuples of distinct elements from a countable set Ω, regarded as a structure with Aut(Ω (n) ) = Sym(Ω) and we show a result of biinterpretability. The material here presented addresses a problem which arises in the context of classification of totally categorical structures. 1

### Proof of Gaifman’s conjecture for relatively categorical abelian groups

"... Abstract. Haim Gaifman conjectured in 1974 that if T is a complete first-order theory which is relatively categorical over its relativisation T P to a predicate P, then every model B of T P can be extended to a model A of T with A P = B. He proved this when T is rigid over P, and it holds also for s ..."

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Abstract. Haim Gaifman conjectured in 1974 that if T is a complete first-order theory which is relatively categorical over its relativisation T P to a predicate P, then every model B of T P can be extended to a model A of T with A P = B. He proved this when T is rigid over P, and it holds also for some cases of relative categoricity in uncountably categorical theories. In what may be the first example going significantly beyond these two types, we show that the conjecture is true when the relatively categorical theory T is the theory of an abelian group with P selecting a subgroup. 1 Gaifman’s conjecture In [5] Haim Gaifman introduced some notions equivalent to the following. Let L be a first-order language, P a 1-ary relation symbol not in L, and L(P) the language which results from adding P to L. Let T be a complete theory in L(P), with the property that if A is any model of T, then the L-reduct