Abstract
Ballester-Bolinches and Guo showed that a finite group G is 2-nilpotent if G satisfies: (1) a Sylow 2-subgroup P of G is quaternion-free and (2) Ω1(P ∩ G′) ≤ Z(P) and N G (P) is 2-nilpotent. In this paper, it is obtained that G is a non-2-nilpotent group of order 16q for an odd prime q satisfying (1) a Sylow 2-subgroup P of G is not quaternion-free and (2) Ω1(P ∩ G′) ≤ Z(P) and N G (P) is 2-nilpotent if and only if q = 3 and G ≅ GL 2(3).
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