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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.

06F15 Ordered groups
20E15 Chains and lattices of subgroups, subnormal subgroups
20E07 Subgroup theorems; subgroup growth
20F60 Ordered groups (group-theoretic aspects)
Full Text: DOI
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