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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:
 2e+16 Chains and lattices of subgroups, subnormal subgroups 2e+11 Quasivarieties and varieties of groups 2e+08 Subgroup theorems; subgroup growth
##### Keywords:
subnormal embeddings; verbal subgroups; varieties of groups
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