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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.
Grunewald, F. +2 more
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Reachability Analysis for Solvable Dynamical Systems
IEEE Transactions on Automatic Control, 2018The reachability problem is one of the most important issues in the verification of hybrid systems. But unfortunately the reachable sets for most of hybrid systems are not computable.
Ting Gan +4 more
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Journal of the London Mathematical Society, 1991
The authors say a group is homogeneous if any isomorphism between two of its finitely generated subgroups is induced by an automorphism. (In model theory there are also other versions of the concept of homogeneity; see the paper of \textit{B. I. Rose} and \textit{R. E. Woodrow} [Z. Math. Logik Grundlagen Math.
Cherlin, Gregory L., Felgner, Ulrich
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The authors say a group is homogeneous if any isomorphism between two of its finitely generated subgroups is induced by an automorphism. (In model theory there are also other versions of the concept of homogeneity; see the paper of \textit{B. I. Rose} and \textit{R. E. Woodrow} [Z. Math. Logik Grundlagen Math.
Cherlin, Gregory L., Felgner, Ulrich
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Journal of Group Theory, 2003
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
LONGOBARDI, Patrizia +2 more
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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
LONGOBARDI, Patrizia +2 more
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Applicable Algebra in Engineering, Communication and Computing, 2003
This paper deals with the problem of determining whether a quintic polynomial is solvable, i.e. whether its roots are expressible by repeated radicals. Every irreducible quintic \(f(x)\) can be transformed, by two applications of Tschirnhausen transformations, into a so-called Brioschi quintic or Brioschi resolvent, with a parameter \(Z\) in the ...
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This paper deals with the problem of determining whether a quintic polynomial is solvable, i.e. whether its roots are expressible by repeated radicals. Every irreducible quintic \(f(x)\) can be transformed, by two applications of Tschirnhausen transformations, into a so-called Brioschi quintic or Brioschi resolvent, with a parameter \(Z\) in the ...
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Mathematics of the USSR-Sbornik, 1989
This article contains complete proofs of the results obtained earlier by the author in his previous short communications which gives description of the orbit structure of the homogeneous flow on compact solvmanifolds. Let G/D be a solvmanifold, where G is a connected solvable Lie group, D- closed subgroup, \(x\in g\) the Lie algebra of G, and \(gD\to ...
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This article contains complete proofs of the results obtained earlier by the author in his previous short communications which gives description of the orbit structure of the homogeneous flow on compact solvmanifolds. Let G/D be a solvmanifold, where G is a connected solvable Lie group, D- closed subgroup, \(x\in g\) the Lie algebra of G, and \(gD\to ...
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Acta Mathematica Sinica, English Series, 2016
A finite group \(G\) is a \(D_{n}\)-group if and only if the number of non-linear irreducible characters of \(G\) is exactly \(n\) less than the number of their different degrees. \textit{Y. Berkovich} et al. [Proc. Am. Math. Soc. 115, No. 4, 955--959 (1992; Zbl 0822.20004)] classified \(D_{0}\)-groups and \textit{Y. Berkovich} and \textit{L. Kazarin} [
Liu, Yang, Lu, Zi Qun
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A finite group \(G\) is a \(D_{n}\)-group if and only if the number of non-linear irreducible characters of \(G\) is exactly \(n\) less than the number of their different degrees. \textit{Y. Berkovich} et al. [Proc. Am. Math. Soc. 115, No. 4, 955--959 (1992; Zbl 0822.20004)] classified \(D_{0}\)-groups and \textit{Y. Berkovich} and \textit{L. Kazarin} [
Liu, Yang, Lu, Zi Qun
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