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The solvable-radical sequence conjecture for finite groups

From papers

Let GG be a finite group, and let R(G)R(G) denote its solvable radical. Suppose (un(x,y))n≥1(u_n(x,y))_{n\geq 1} is a sequence of group words in two variables.

Solvable-radical sequence conjecture. There exists a sequence un(x,y)u_n(x,y) such that R(G)R(G) coincides with the set of elements y∈Gy\in G having the property that, for every x∈Gx\in G, there is an nn for which

un(x,y)=1.u_n(x,y)=1.

Here the sequence is intended to characterize the solvable radical uniformly for each finite group through the vanishing of one of its words for every choice of xx.

The source states this as a conjecture whose validity would imply the finite-group characterization S(G)=R(G)S(G)=R(G), where S(G)S(G) is the set of radical elements. No resolution is supplied in the given text.

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Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The solvable-radical sequence conjecture for finite groups

    Let GG be a finite group. For a sequence u→=u→(x,y)\overrightarrow{u}=\overrightarrow{u}(x,y), write G(u→)G(\overrightarrow{u}) for the subgroup associated with the sequence and let the solvable radical of GG be its largest solvable normal subgroup. Solvable-radical sequence conjecture. There is a sequence u→=u→(x,y)\overrightarrow{u}=\overrightarrow{u}(x,y) such that, for every finite group GG, the solvable radical of GG coincides with G(u→)G(\overrightarrow{u}). This seeks a uniform Engel-like description of solvable radicals in finite groups; the supplied text does not state whether the conjecture has been resolved.

    source: Tatiana Bandman, Mikhail Borovoi, Fritz Grunewald, Boris Kunyavskii and Eugene Plotkin, “Engel-like characterization of radicals in finite dimensional Lie algebras and finite groups”, arXiv:math/0411463 (2004).

Sources & referencesView supporting material

Primary source

R. Guralnick, B. Kunyavskii, E. Plotkin and A. Shalev, “Thompson-like characterization of the solvable radical”, arXiv:math/0504176 (2005).

engel-conditionsfinite-groupsgroup-wordssolvable-radical

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