A description of Baer–Suzuki type of the solvable radical of a finite group
نویسندگان
چکیده
We obtain the following characterization of the solvable radicalR(G) of any finite group G: R(G) coincides with the collection of all g ∈ G such that for any 3 elements a1, a2, a3 ∈ G the subgroup generated by the elements g, aiga 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). © 2008 Elsevier B.V. All rights reserved.
منابع مشابه
From Thompson to Baer–suzuki: a Sharp Characterization of the Solvable Radical
We prove that an element g of prime order > 3 belongs to the solvable radical R(G) of a finite (or, more generally, a linear) group if and only if for every x ∈ G the subgroup generated by g, xgx is solvable. This theorem implies that a finite (or a linear) group G is solvable if and only if in each conjugacy class of G every two elements generate a solvable subgroup.
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