HSP\(\neq SHPS\) for metabelian groups, and related results
type: Item
instance of: scholarly article
title: HSP\(\neq SHPS\) for metabelian groups, and related results
zbMATH Open document ID: 0627.08006
DOI: 10.1007/BF01211835
author: George M. Bergman
published in: Algebra Universalis
publication date: 1989-01-01T00:00:00Z
review text: Consider the standard operators on classes of algebras, H (homomorphic images), S (subalgebras) and P (products), and the further operator \(P_ f\) of finite products. It is shown by counterexamples that HSP\(\neq SHPS\) and \(HSP_ f\neq SHP_ fS\) for metabelian groups (groups satisfying \(G''=\{e\})\), and that \(HSP_ f\not\subseteq SHPS\) for solvable groups (in fact, for finite groups satisfying \((G',G'')=\{e\})\). From the first two inequalities and some easier examples, it is deduced that the partially ordered semigroups of operators on metabelian groups generated by \(\{\) H,S,P\(\}\) and by \(\{H,S,P_ f\}\) are as in the ``standard'' 18- element diagram. [Cf. \textit{E. Nelson}, Can. J. Math. 17, 764-768 (1967; Zbl 0149.260), \textit{P. M. Neumann}, Bull. Am. Math. Soc. 76, 1067-1069 (1970; Zbl 0209.321), \textit{D. Pigozzi}, Algebra Univers. 2, 346-353 (1972; Zbl 0272.08006).]
Mathematics Subject Classification ID: 08B25
Mathematics Subject Classification ID: 20E10
Mathematics Subject Classification ID: 20F16
zbMATH DE Number: 4019136
zbMATH Keywords: finite products
zbMATH Keywords: metabelian groups
zbMATH Keywords: solvable groups
zbMATH Keywords: partially ordered semigroups of operators on metabelian groups
MaRDI profile type: Publication
cites work: The Birkhoff theorem for varieties of finite algebras
cites work: The standard semigroup of operators of a variety
cites work: Finiteness of Semigroups of Operators in Universal Algebra
cites work: The inequality of ${\text{SQPS}}$ and ${\text{QSP}}$ as operators on classes of groups
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description: scientific article; zbMATH DE number 4019136