TY - JOUR
T1 - Universal theories for rigid soluble groups
AU - Myasnikov, A. G.
AU - Romanovskii, N. S.
PY - 2012/1
Y1 - 2012/1
N2 - 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 > ... > G p > G p+1 = 1, whose quotients G i/G i+1 are Abelian and are torsion free when treated as 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 ∀-theories for these groups satisfy the inclusions A(F) ⊇ A(G) ⊇ A(W). We construct an ∃-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(α 1,..., α p). The latter group factors into a semidirect product of Abelian groups A 1A 2... 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 α i over a ring 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.
AB - 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 > ... > G p > G p+1 = 1, whose quotients G i/G i+1 are Abelian and are torsion free when treated as 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 ∀-theories for these groups satisfy the inclusions A(F) ⊇ A(G) ⊇ A(W). We construct an ∃-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(α 1,..., α p). The latter group factors into a semidirect product of Abelian groups A 1A 2... 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 α i over a ring 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.
KW - decidable theory
KW - p-rigid group
KW - universal theory of group
UR - http://www.scopus.com/inward/record.url?scp=84858753364&partnerID=8YFLogxK
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U2 - 10.1007/s10469-012-9164-y
DO - 10.1007/s10469-012-9164-y
M3 - Article
AN - SCOPUS:84858753364
SN - 0002-5232
VL - 50
SP - 539
EP - 552
JO - Algebra and Logic
JF - Algebra and Logic
IS - 6
ER -