Universal theories for rigid soluble groups

A. G. Myasnikov ; N. S. Romanovskii

Algebra i logika, Tome 50 (2011) no. 6, pp. 802-821.

Résumé

A group is said to be $p$-rigid, where $p$ is a natural number, if it has a normal series of the form $$ G=G_1>G_2>\dots>G_p>G_{p+1}=1, $$ whose quotients $G_i/G_{i+1}$ are Abelian and are torsion free when treated as $\mathbb Z[G/G_i]$-modules. Examples of rigid groups are free soluble groups. We point out a recursive system of universal axioms distinguishing $p$-rigid groups in the class of $p$-soluble groups. It is proved that if $F$ is a free $p$-soluble group, $G$ is an arbitrary $p$-rigid group, and $W$ is an iterated wreath product of $p$ infinite cyclic groups, then $\forall$-theories for these groups satisfy the inclusions $$ \mathcal A(F)\supseteq\mathcal A(G)\supseteq\mathcal A(W). $$ We construct an $\exists$-axiom distinguishing among $p$-rigid groups those that are universally equivalent to $W$. An arbitrary $p$-rigid group embeds in a divisible decomposed $p$-rigid group $M=M(\alpha_ 1,\dots,\alpha_ p)$. The latter group factors into a semidirect product of Abelian groups $A_1A_2\dots A_p$, in which case every quotient $M_i/M_{i+1}$ of its rigid series is isomorphic to $A_i$ and is a divisible module of rank $\alpha_i$ over a ring $\mathbb Z[M/M_i]$. We specify a recursive system of axioms distinguishing among $M$-groups those that are Muniversally equivalent to $M$. As a consequence, it is stated that the universal theory of $M$ with constants in $M$ is decidable. By contrast, the universal theory of $W$ with constants is undecidable.

Détail

Mots-clés : $p$-rigid group
Keywords: universal theory of group, decidable theory.

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