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HSP ≠ SHPS for metabelian groups, and related results

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Abstract

For the standard operators on classes of algebras,H (homomorphic images),S (subalgebras) andP (products), and the further operatorP f of finite products, it is shown by counterexamples thatHSP ≠SHPS andHSP f ≠SHP fS for metabelian groups (groups satisfyingG″={e}) and thatHSP f SHPS for solvable groups (in fact, for finite groups satisfying (G′, G″)= {e}). From the first two inequalities and some easier examples, it follows 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.

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Authors and Affiliations

  1. University of California, 94720, Berkeley, CA, USA

    George M. Bergman

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  1. George M. Bergman
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In Memory of Evelyn Nelson

This work was done while the author was partly supported by NSF contracts MCS 82-02632 and DMS 85-02330.

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Bergman, G.M. HSP ≠ SHPS for metabelian groups, and related results. Algebra Universalis 26, 267–283 (1989). https://doi.org/10.1007/BF01211835

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  • DOI: https://doi.org/10.1007/BF01211835

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