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The automorphism groups of generalized McLain groups. (English) Zbl 0918.20026
Holland, W. Charles (ed.), Ordered groups and infinite permutation groups. Partially based on the conference, Luminy, France, Summer 1993. Dordrecht: Kluwer Academic Publishers. Math. Appl., Dordr. 354, 97-120 (1996).
The paper concerns certain groups $$G(R,S)$$, built from an arbitrary ring $$R$$ without zero-divisors and an arbitrary partially ordered set $$(S,\leq)$$ which is unbounded and locally linear (that is, if $$\alpha,\beta\in S$$ with $$\alpha<\beta$$ then $$\{x:\alpha<x<\beta\}$$ is linearly ordered). These generalise groups constructed by McLain from the finite field $$\mathbb{F}_p$$ and the rational line. The structure of the automorphism group of $$G(R,S)$$ was described in earlier work by the authors. Here they show that if two such groups $$G(R,S)$$ and $$G(R',S')$$ have isomorphic automorphism groups, then the groups are isomorphic, provided each automorphism of $$R$$ is determined by its action on the units $$U(R)$$ (and likewise for $$R'$$). The paper contains a number of intermediate results characterising certain abelian subgroups of $$G(R,S)$$.
For the entire collection see [Zbl 0899.00027].

##### MSC:
 20F28 Automorphism groups of groups 20E07 Subgroup theorems; subgroup growth 20H25 Other matrix groups over rings