A subgroup H of a finite group G is said to be Hall normally embedded in G if there is a normal subgroup N of G such that H is a Hall subgroup of N. Adolfo Ballester-Bolinches and ShouHong Qiao [(Arch. Math. 102 (2014), 109–111] proved that a group G has a Hall normally embedded subgroup of order |B| for each subgroup B of G if and only if the nilpotent residual of G is cyclic of square-free order. This is the answer to a problem posed by Li and Liu [J. Algebra 388 (2013), 1–9]. We prove that the nilpotent residual of a finite group G is cyclic of square-free order if for each p ∈ π(G), and for each subgroup B with the Sylow p-subgroups of order p, there exists an S-permutable subgroup with a Hall subgroup of order |B|.
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