Dixon, Martyn R.; Fournelle, Thomas A. Wreath products indexed by partially ordered sets. (English) Zbl 0591.20031 Rocky Mt. J. Math. 16, 7-15 (1986). In [Proc. Camb. Philos. Soc. 58, 170-184 (1962; Zbl 0109.013)], P. Hall introduced a generalized wreath product, proved a number of results about this structure and used it to provide numerous interesting examples. These definitions were further generalized by W. C. Holland [J. Algebra 13, 152-172 (1969; Zbl 0186.038)] and the authors [Arch. Math. 43, 193-207 (1984; Zbl 0532.20017) and Compos. Math. 52, 355-372 (1984; Zbl 0552.20014)]. In these cases the index set was always totally ordered. Here the authors generalize it further to partially ordered sets. Many of Philip Hall’s results are extended to this more general situation. In particular some results about automorphisms are proved, especially those connected with order automorphisms of the partially ordered set. There are also several results on residual finiteness, finite conjugacy and the closure properties of classes of groups. The existence of a number of groups with unusual properties is shown. An example of this is that a group is constructed for every prime p which is infinite, locally finite, residually finite, FC, neither Hopfian nor co-Hopfian, of exponent \(p^ 2\), nilpotent, metabelian and indecomposable. There is probably a lot more work of interest to be done with these groups. Reviewer: J.D.P.Meldrum Cited in 3 Documents MSC: 20E22 Extensions, wreath products, and other compositions of groups 20E25 Local properties of groups 20E26 Residual properties and generalizations; residually finite groups 20E34 General structure theorems for groups 20F14 Derived series, central series, and generalizations for groups 20E36 Automorphisms of infinite groups 20F24 FC-groups and their generalizations Keywords:generalized wreath product; partially ordered sets; automorphisms; residual finiteness; locally finite; residually finite; FC; Hopfian Citations:Zbl 0541.20017; Zbl 0109.013; Zbl 0186.038; Zbl 0532.20017; Zbl 0552.20014 PDFBibTeX XMLCite \textit{M. R. Dixon} and \textit{T. A. Fournelle}, Rocky Mt. J. Math. 16, 7--15 (1986; Zbl 0591.20031) Full Text: DOI