TY - JOUR

T1 - Divisible Rigid Groups. III. Homogeneity and Quantifier Elimination

AU - Romanovskii, N. S.

PY - 2019/1/15

Y1 - 2019/1/15

N2 - A group G is said to be rigid if it contains a normal series G = G 1 > G 2 >.. > G m > G m+1 = 1, whose quotients G i /G i+1 are Abelian and, treated as right ℤ[G/G i ]-modules, are torsion-free. A rigid group G is divisible if elements of the quotient G i /G i+1 are divisible by nonzero elements of the ring ℤ[G/G i ]. 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 ℵ 0 -homogeneous and that the theory of divisible m-rigid groups admits quantifier elimination down to a Boolean combination of ∃-formulas.

AB - A group G is said to be rigid if it contains a normal series G = G 1 > G 2 >.. > G m > G m+1 = 1, whose quotients G i /G i+1 are Abelian and, treated as right ℤ[G/G i ]-modules, are torsion-free. A rigid group G is divisible if elements of the quotient G i /G i+1 are divisible by nonzero elements of the ring ℤ[G/G i ]. 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 ℵ 0 -homogeneous and that the theory of divisible m-rigid groups admits quantifier elimination down to a Boolean combination of ∃-formulas.

KW - divisible group

KW - quantifier elimination

KW - rigid group

KW - strongly ℵ -homogeneous group

KW - strongly (0)-homogeneous group

UR - http://www.scopus.com/inward/record.url?scp=85063814703&partnerID=8YFLogxK

U2 - 10.1007/s10469-019-09518-2

DO - 10.1007/s10469-019-09518-2

M3 - Article

AN - SCOPUS:85063814703

VL - 57

SP - 478

EP - 489

JO - Algebra and Logic

JF - Algebra and Logic

SN - 0002-5232

IS - 6

ER -