Contractions of infra-invariant systems of subgroups.

*(English. Russian original)*Zbl 1241.06015
Algebra Logic 48, No. 5, 344-356 (2009); translation from Algebra Logika 48, No. 5, 606-627 (2009).

Summary: We create a method which allows an arbitrary group \(G\) with an infra-invariant system \(\mathcal L(G)\) of subgroups to be embedded in a group \(G^*\) with an infra-invariant system \(\mathcal L(G^*)\) of subgroups so that \(G_\alpha^*\cap G\in\mathcal L(G)\) for every subgroup \(G_\alpha^*\in\mathcal L(G^*)\) and each factor \(B/A\) of a jump of subgroups in \(\mathcal L(G^*)\) is isomorphic to a factor of a jump in \(\mathcal L(G)\) or to any specified group \(H\). Using this method, we state new results on right-ordered groups. In particular, it is proved that every Conrad right-ordered group is embedded with preservation of order in a Conrad right-ordered group of Hahn type (i.e., a right-ordered group whose factors of jumps of convex subgroups are order-isomorphic to the additive group \(\mathbb R\)); every right-ordered Smirnov group is embedded in a right-ordered Smirnov group of Hahn type; a new proof is given for the Holland-McCleary theorem on embedding every linearly ordered group in a linearly ordered group of Hahn type.

##### MSC:

06F15 | Ordered groups |

20E15 | Chains and lattices of subgroups, subnormal subgroups |

20E07 | Subgroup theorems; subgroup growth |

20F60 | Ordered groups (group-theoretic aspects) |

##### Keywords:

infra-invariant systems of subgroups; right-ordered groups; linearly ordered groups; linearly ordered group of Hahn type
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\textit{V. M. Kopytov}, Algebra Logic 48, No. 5, 344--356 (2009; Zbl 1241.06015); translation from Algebra Logika 48, No. 5, 606--627 (2009)

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##### References:

[1] | M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of Group Theory [in Russian], Nauka, Moscow (1984). · Zbl 0884.20001 |

[2] | V. M. Kopytov, Lattice-Ordered Groups [in Russian], Nauka, Moscow (1984). · Zbl 0567.06011 |

[3] | V. M. Kopytov and N. Ya. Medvedev, Right-Ordered Groups, Sib. School Alg. Log. [in Russian], Nauch. Kniga, Novosibirsk (1996). |

[4] | W. C. Holland and S. H. McCleary, ”Wreath products of ordered permutation groups,” Pac. J. Math., 31, 703-716 (1969). · Zbl 0206.31804 · doi:10.2140/pjm.1969.31.703 |

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