Soluble Groups and Varieties of $l$-Groups

N. Ya. Medvedev

Algebra i logika, Tome 44 (2005) no. 3, pp. 355-367

Résumé

A sufficient condition is given under which factors of a system of normal convex subgroups of a linearly ordered (l. o. ) group are Abelian. Also, a sufficient condition is specified subject to which factors of a system of normal convex subgroups of an l. o. group are contained in a group variety $\mathcal V$. In particular, for every soluble l. o. group $G$ of solubility index $n$, $n\geqslant2$, factors of a system of normal convex subgroups are soluble l. o. groups of solubility index at most $n-1$. It is proved that the variety $\mathcal R$ of all lattice-ordered groups, approximable by linearly ordered groups, does not coincide with a variety generated by all soluble l. o. groups. It is shown that if $\mathcal V$ is any $o$-approximable variety of $l$-groups, and if every identity in the group signature is not identically true in $\mathcal V$, then $\mathcal V$ contains free l. o. groups.

Détail

Keywords: variety of $l$-groups
Mots-clés : soluble group.