Divisible rigid groups. II. Stability, saturation, and elementary submodels

A. G. Myasnikov ; N. S. Romanovskii

Algebra i logika, Tome 57 (2018) no. 1, pp. 43-56

Résumé

A group $G$ is said to be rigid if it contains a normal series $$ G=G_1>G_2>\dots>G_m>G_{m+1}=1, $$ 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 divisible if elements of the quotient $G_i/G_{i+1}$ are divisible by nonzero elements of the ring $\mathbb Z[G/G_i]$. Every rigid group is embedded in a divisible one. Previously, it was stated that the theory $\mathfrak T_m$ of divisible $m$-rigid groups is complete. Here, it is proved that this theory is $\omega$-stable. Furthermore, we describe saturated models, study elementary submodels of an arbitrary model, and find a representation for a countable saturated model in the form of a limit group in the Fraïssé system of all finitely generated $m$-rigid groups. Also, it is proved that the theory $\mathfrak T_m$ admits quantifier elimination down to a Boolean combination of $\forall\exists$-formulas.

Détail

Mots-clés : divisible rigid group, saturation
Keywords: theory, model, stability, $\forall\exists$-formula.