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CHARACTERIZATION OF SOLVABLE GROUPS AND SOLVABLE RADICAL
International Journal of Algebra and Computation, 2013We give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were obtained over the past decade. We also discuss generalizations of these results to some classes of infinite groups and their analogues for Lie algebras. Some open problems are discussed as well.
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A group \(G\) is called \(R^*\)-group if for all \(n>0\) and elements \(g\) and \(x_1,\dots,x_n\) the equation \(g^{x_1 }\cdots g^{x_n }=1\) implies \(g=1\). The following results are proved: (1) if \(G\) is an Abelian-by-nilpotent as well as nilpotent-by-Abelian \(R^*\)-group, then every partial order on \(G\) can be extented to a linear order; (2) if
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