# zbMATH — the first resource for mathematics

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)
Full Text:
##### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.