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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].

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