TY - JOUR

T1 - A COMMUTATOR DESCRIPTION OF THE SOLVABLE RADICAL OF A FINITE GROUP

AU - NIKOLAI, GORDEEV

AU - FRITZ, GRUNEWALD

AU - Kunyavskii, B.

AU - EUGENE, PLOTKIN

PY - 2007

Y1 - 2007

N2 - 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 g ∈ G
such that for any k elements a1, a2, . . . , ak ∈ G the subgroup generated
by the elements g, aiga−1
i
, 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 g ∈ G
such that for any ℓ elements b1, b2, . . . , bℓ ∈ G the subgroup generated
by the commutators [g, bi
], 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 every 8 conjugate elements of G generate a solvable
subgroup

AB - 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 g ∈ G
such that for any k elements a1, a2, . . . , ak ∈ G the subgroup generated
by the elements g, aiga−1
i
, 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 g ∈ G
such that for any ℓ elements b1, b2, . . . , bℓ ∈ G the subgroup generated
by the commutators [g, bi
], 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 every 8 conjugate elements of G generate a solvable
subgroup

UR - http://arxiv.org/pdf/math/0610983.pdf

M3 - Article

SP - 85

EP - 120

JO - Groups, Geometry, and Dynamics 2

JF - Groups, Geometry, and Dynamics 2

ER -