## 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

Zbl 0182.35502
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