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Adolfo Ballester-Bolinches, Shouhong Qiao

On a problem posed by S. Li and J. Liu

Arch. Math., 102, 109-111 (2014)

Abstract: 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 . The aim of this note is to prove that a group G has
a Hall normally embedded subgroup of order |B| for each subgroup B of
G if and only if G is soluble with nilpotent residual cyclic of square-free
order. This is the answer to a problem posed by Li and Liu (J. Algebra
388:1–9, 2013).

MSC: 20D10, 20D20
Keywords: Finite group, soluble group, Hall subgroups