Contractions of infrainvariant systems of subgroups

V. M. Kopytov

Algebra i logika, Tome 48 (2009) no. 5, pp. 606-627

Résumé

We create a method which allows an arbitrary group $G$ with an infrainvariant system $\mathcal L(G)$ of subgroups to be embedded in a group $G^*$ with an infrainvariant 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.

Détail

Keywords: infrainvariant system of subgroups, right-ordered group, linearly ordered group, linearly ordered group of Hahn type.