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McLain groups over arbitrary rings and orderings. (English) Zbl 0839.20050
D. H. McLain [Math. Proc. Camb. Philos. Soc. 50, 641-642 (1954; Zbl 0056.02201)] applied arguments of linear algebra to establish the existence of characteristically simple, locally finite \(p\)-groups. These McLain groups \(G(F_p,\mathbb{Q})\) of upper-triangular \(\mathbb{Q}\times\mathbb{Q}\) matrices are constructed from the linearly ordered set \((\mathbb{Q},\leq)\) of rational numbers and the field \(F_p\) of \(p\) elements. \(G(F_p,S)\) has since been studied for other linear orderings \(S\) by J. E. Roseblade [Math. Z. 82, 267-282 (1963; Zbl 0115.25201)] and by J. S. Wilson [Math. Proc. Camb. Philos. Soc. 86, 193-197 (1979; Zbl 0416.20025)].
Here there is a much more general point of view. Let \(S\) be any partially ordered set, and \(R\) any ring with \(1\neq 0\). A generalized McLain group \(G(R,S)\) is constructed as before. An arbitrary \(g\in G(R,S)\) can be uniquely expressed as \(g=1+\sum_{\alpha<\beta}a_{\alpha\beta}e_{\alpha\beta}\), with \(a_{\alpha\beta}\in R\) and \(a_{\alpha \beta}=0\) for almost all \(\alpha,\beta\in S\). Multiplication is induced by matrix multiplication of \(S\times S\) matrix units \(e_{\alpha\beta}\). Relations are established between group theoretic properties of \(G=G(R,S)\) and properties of \(R\) and \(S\). Suppose here that \(S\) is locally linear (i.e., for all \(\alpha,\beta\in S\), \(\{\sigma:\alpha<\sigma<\beta\}\) is linearly ordered), unbounded, and connected, and that \(R\) has no zero divisors.
Theorem. \(G(R,S)\cong G(R',S')\) if and only if either \(R\cong R'\) and \(S\cong S'\), or \(R\) is anti-isomorphic to \(R'\) and \(S\) to \(S'\). Theorem. If \(S\) is loop-free and all units of \(R\) are central, then \[ \operatorname{Aut} G\cong(((\text{Linn }G\times U(R)^{|S|})\times\operatorname{Aut} R)\times\operatorname{Aut} S)\times\langle\tau\rangle. \] Here \(\text{Linn}(G)\times U(R)^{|S|}\) is the semidirect product of the locally inner automorphism group of \(G\) with the full direct product of \(|S|\) copies of the group of units of \(R\), and \(\tau\) has order 1 or 2. Theorem. Suppose \(S\) has no anti-automorphisms. Then \(G(R,S)\) is characteristically simple if and only if \(S\) is feebly 2-homogeneous (i.e., for all \(\alpha<\beta\) and \(\gamma<\delta\) in \(S\) there exists \(f\in \operatorname{Aut} S\) such that \(\gamma\leq\alpha f<\beta f\leq\delta)\).
The extra freedom stemming from not requiring \(S\) to be linearly ordered permits the construction of: (1) For each prime \(p\), \(2^{\aleph_0}\) countable characteristically simple locally finite \(p\)-groups \(G(F_p,S)\). (2) \(2^{\aleph_0}\) countable locally finite 2-groups \(G\) such that \(\operatorname{Aut} G=\text{Linn }G\).

20F19 Generalizations of solvable and nilpotent groups
20F28 Automorphism groups of groups
20H25 Other matrix groups over rings
20F50 Periodic groups; locally finite groups
20E32 Simple groups
20E25 Local properties of groups
Full Text: DOI
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