A subgroup H of a group G is called Q-permutable in G if there exists a subgroup B of G such that (1) G = HB and (2) if H 1 is a maximal subgroup of H containing H QG , then H 1 B = BH 1 < G, where H QG is the largest permutable subgroup of G contained in H. In this paper, we prove the following statement: Let \( \mathcal{F} \) be a saturated formation containing \( \mathcal{U} \) and let G be a group with a normal subgroup H such that \( {{G} \left/ {H} \right.} \in \mathcal{F} \). If every maximal subgroup of every noncyclic Sylow subgroup of F*(H) having no supersolvable supplement in G is Q-permutable in G, then \( G \in \mathcal{F} \).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 11, pp. 1534–1543, November, 2011.
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Pu, Z., Miao, L. Q-Permutable subgroups of finite groups. Ukr Math J 63, 1745–1755 (2012). https://doi.org/10.1007/s11253-012-0610-8
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DOI: https://doi.org/10.1007/s11253-012-0610-8