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Characterization of \(r\)-solvable groups. (English. Russian original) Zbl 0956.20006
Sib. Math. J. 41, No. 1, 180-187 (2000); translation from Sib. Mat. Zh. 41, No. 1, 214-223 (2000).
The following theorem is proven which generalizes a result by G. Glauberman [Ill. J. Math. 12, 76-98 (1968; Zbl 0182.35502)]: Let \(G\) be a finite \(K\)-group and let \(r\) be a prime divisor of the order of \(G\). Then \(G\) is \(r\)-soluble if and only if every pair of elements in \(G\) generates an \(r\)-soluble subgroup.
Recall that a group \(G\) is called \(K\)-group if all its composition factors are known simple groups. First, the author proves that a minimal counterexample must be a finite simple group. Then the author proves that every known simple group is either \(r\)-soluble or contains two \(r\)-elements which generate a non-\(r\)-soluble group.
MSC:
20D05 Finite simple groups and their classification
20D06 Simple groups: alternating groups and groups of Lie type
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
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