A commutator description of the solvable radical of a finite group

Details A commutator description of the solvable radical of a finite group A commutator description of the solvable radical of a finite group

2006-10-31

arxiv.org/abs/math/0610983v2

Groups, Geometry, and Dynamics 2 (2008), No. 1, 85-120

Mathematics

Group Theory

43 pages

Scientific paper

We are looking for the smallest integer k>1 providing the following characterization of the solvable radical R(G) of any finite group G: R(G) coincides with the collection of all g such that for any k elements a_1,a_2,...,a_k the subgroup generated by the elements g, a_iga_i^{-1}, i=1,...,k, is solvable. We consider a similar problem of finding the smallest integer l>1 with the property that R(G) coincides with the collection of all g such that for any l elements b_1,b_2,...,b_l the subgroup generated by the commutators [g,b_i], i=1,...,l, is solvable. Conjecturally, k=l=3. We prove that both k and l are at most 7. In particular, this means that a finite group G is solvable if and only if in each conjugacy class of G every 8 elements generate a solvable subgroup.

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