Generic types and generic elements in divisible rigid groups

A. G. Myasnikov ; N. S. Romanovskii

Algebra i logika, Tome 62 (2023) no. 1, pp. 102-113

Résumé

A group $G$ is said to be $m$-rigid if it contains a normal series of the form $$G=G_1>G_2>\ldots>G_m>G_{m+1}=1,$$ whose quotients $G_i/G_{i+1}$ are Abelian and, treated as (right) ${\mathbb{Z}}[G/G_i]$-modules, are torsion-free. A rigid group $G$ is said to be divisible if elements of the quotient $\rho_i(G)/\rho_{i+1}(G)$ are divisible by nonzero elements of the ring ${\mathbb{Z}}[G/\rho_i(G)]$. Previously, it was proved that the theory of divisible $m$-rigid groups is complete and $\omega$-stable. In the present paper, we give an algebraic description of elements and types that are generic over a divisible $m$-rigid group $G$.

Détail

Mots-clés : divisible $m$-rigid group
Keywords: generic type, generic element.