## Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 478-489 |

Number of pages | 12 |

Journal | Algebra and Logic |

Volume | 57 |

Issue number | 6 |

DOIs | |

Publication status | Published - 15 Jan 2019 |

## Keywords

- divisible group
- quantifier elimination
- rigid group
- strongly ℵ -homogeneous group
- strongly (0)-homogeneous group