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Universal theories for rigid soluble groups
A. G. Myasnikova,
N. S. Romanovskiibc a Schaefer School of Engineering and Science, Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ, USA
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
c Novosibirsk State University, Novosibirsk, Russia
Abstract:
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.
Keywords:
$p$-rigid group, universal theory of group, decidable theory.
UDC:
512.54.05 Received: 01.03.2011