Characterization of \(r\)-solvable groups
The following theorem is proven which generalizes a result by \textit{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.
scientific article; zbMATH DE number 1534387