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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 $$\operatorname{Aut} G=\text{Linn }G$$.

##### MSC:
 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
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##### References:
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