Some questions of Edjvet and Pride about infinite groups
The reviewer has introduced a quasi-order \(\preccurlyeq\) on the class of groups, with the aim of making precise, vague notions of when one group is ''larger'' than another. [See \textit{S. J. Pride} in Word Problems II, 299-335 (1980; Zbl 0438.20023), \textit{M. Edjvet} and \textit{S. J. Pride}, in Groups-Korea 1983, Lect. Notes Math. 1098, 29-54 (1984; Zbl 0566.20014).] Write \(G\simeq H\) if \(G\preccurlyeq H\) and \(H\preccurlyeq G\), and let [G] denote the \(\simeq\)-equivalence class containing G. Then \(\preccurlyeq\) induces a partial order (also denoted \(\preccurlyeq)\) on the equivalence classes. The ideal Id[G] is the partially ordered set consisting of all [H]\(\preccurlyeq [G]\). We say that G has height n if the partially ordered set Id[G] has height n. In the two papers mentioned above several questions concerning the partial order \(\preccurlyeq\) were raised. Several of these questions are answered in the paper under review. The author shows by example that it is not necessarily true that if G has finite height then all maximal chains in Id[G] have the same length. Also, it need not be the case that if G is a finitely generated group of finite height then Id[G] is finite. It was suggested in the article of Edjvet and Pride mentioned above that the finitely generated groups of height \(\leq 1\) are precisely the groups which are finite-by \({\mathcal D}_ 2\)-by finite (here \({\mathcal D}_ 2\) is the class of groups in which every subnormal subgroup has finite index). The author shows that this is not so. Further results are concerned with SQ-universal groups. In particular, a finitely generated SQ-universal group is constructed which is of height 3. (This leads to the inconsequential, but nevertheless intriguing, question as to whether a finitely generated SQ-universal group can have height 2 [such a group cannot have height 0 or 1].) The author's construction make fairly elaborate use of wreath products. The paper is very clearly written.
scientific article; zbMATH DE number 3964162