Wreath products indexed by partially ordered sets (Q1074714)

In [Proc. Camb. Philos. Soc. 58, 170-184 (1962; Zbl 0109.013)], \textit{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 \textit{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.