A description of Baer–Suzuki type of the solvable radical of a finite group (original) (raw)

2009, Journal of Pure and Applied Algebra

We obtain the following characterization of the solvable radical R(G) of any finite group G: R(G) coincides with the collection of all g ∈ G such that for any 3 elements a 1 , a 2 , a 3 ∈ G the subgroup generated by the elements g, a i ga −1 i , i = 1, 2, 3, is solvable. In particular, this means that a finite group G is solvable if and only if in each conjugacy class of G every 4 elements generate a solvable subgroup. The latter result also follows from a theorem of P. Flavell on {2, 3}-elements in the solvable radical of a finite group (which does not use the classification of finite simple groups).

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