Assume that G is a finite group, π(G) = {s} ∪ σ, s > 2, Σ is a set of Sylow σ-subgroups in which one subgroup is taken for each pi 2 σ, and R(G) is the largest normal soluble subgroup in G (soluble radical of G). Moreover, suppose that each Sylow pi-subgroup Gpi 𝜖 Σ normalizes the s-subgroup T(i) ≠ 1 of the group G. In this case, we establish the conditions under which s divides |R(G)|.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 3, pp. 326–339, March, 2020.
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Bashun, S.Y., Palchik, E.M. On Soluble Radicals of Finite Groups. Ukr Math J 72, 370–385 (2020). https://doi.org/10.1007/s11253-020-01788-9
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DOI: https://doi.org/10.1007/s11253-020-01788-9