×

zbMATH — the first resource for mathematics

Subnormal embedding theorems for groups. (English) Zbl 1030.20016
For any non-trivial word set \(V\subseteq F_\infty\) an arbitrary group \(H\) is subnormally embeddable into some group \(G\) if the corresponding image of \(H\) lies in the verbal subgroup \(V(G)\) of \(G\). If \(H\) is infinite, \(G\) can be chosen to be of the same cardinality as \(H\). Moreover, if \(H\) is countable, \(G\) can be chosen to be a two-generator group. This strengthens the theorem of G. Higman, B. H. Neumann and H. Neumann [J. Lond. Math. Soc. 24, 247-254 (1950; Zbl 0034.30101)] on embeddability of countable groups into two-generator groups, as well as later results of B. H. Neumann and H. Neumann [J. Lond. Math. Soc. 34, 465-479 (1959; Zbl 0102.26401)] on the embedding of \(H\) into the second derived subgroup of \(G\). The subnormality of embedding cannot in general be replaced by normality [see also H. Heineken, V. H. Mikaelian, J. Math. Sci., New York 100, No. 1, 1915-1924 (2000; Zbl 0953.20017)].

MSC:
20E15 Chains and lattices of subgroups, subnormal subgroups
20E10 Quasivarieties and varieties of groups
20E07 Subgroup theorems; subgroup growth
PDF BibTeX XML Cite
Full Text: DOI arXiv