Divisible Rigid Groups. III. Homogeneity and Quantifier Elimination
Divisible Rigid Groups. III. Homogeneity and Quantifier Elimination
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可分刚性群。
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发表时间：
2019
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通讯作者：
N. Romanovskii
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作者：
N. Romanovskii
一个群G称为刚性群，如果它包含一个正规级数G = G1 > G2 >。. . > Gm > Gm+1 = 1，它的同分块Gi/Gi+1是阿贝尔模，并且作为右[G/Gi]-模，是挠自由的.一个刚性群G是可整除的，如果商Gi/Gi+1的元素可被环[G/Gi]的非零元素整除。每一个刚性的群都嵌入到一个可分的群中。我们的主要结果是如下定理。设G是可分刚性群.那么群G的元素的等长元组的n-型的重合意味着这些元组通过G的一个自同构是共轭的。作为推论，我们指出，可分刚性群是强n-0-齐次的，可分m-刚性群的理论允许量词消去到n-公式的布尔组合。
A group G is said to be rigid if it contains a normal series G = G1 > G2 > . . . > Gm > Gm+1 = 1, whose quotients Gi/Gi+1 are Abelian and, treated as right ℤ[G/Gi]-modules, are torsion-free. A rigid group G is divisible if elements of the quotient Gi/Gi+1 are divisible by nonzero elements of the ring ℤ[G/Gi]. Every rigid group is embedded in a divisible one. Our main result is the theorem which reads as follows. Let G be a divisible rigid group. Then the coincidence of ∃-types of same-length tuples of elements of the group G implies that these tuples are conjugate via an automorphism of G. As corollaries we state that divisible rigid groups are strongly ℵ0-homogeneous and that the theory of divisible m-rigid groups admits quantifier elimination down to a Boolean combination of ∃-formulas.