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

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https://doi.org/10.4171/GGD/32

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##### Citation

Gordeev, N., Grunewald, F., Kunyavskii, B., & Plotkin, E. (2008). A commutator
description of the solvable radical of a finite group.* Groups, Geometry, and Dynamics,*
*2*(1), 85-120. doi:10.4171/GGD/32.

Cite as: https://hdl.handle.net/21.11116/0000-0006-9558-F

##### Abstract

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.

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.