Abstract
A subgroup H of a group G is called weakly s-permutable in G if there is a subnormal subgroup T of G such that G = HT and H ∩ T ≤ H sG , where H sG is the maximal s-permutable subgroup of G contained in H. We improve a nice result of Skiba to get the following
Theorem. Let ℱ be a saturated formation containing the class of all supersoluble groups
and let G be a group with E a normal subgroup of G such that G/E ∈ ℱ. Suppose that each noncyclic Sylow p-subgroup P of F*(E) has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| are weakly s-permutable in G for all p ∈ π(F*(E)); moreover, we suppose that every cyclic subgroup of P of order 4 is weakly s-permutable in G if P is a nonabelian 2-group and |D| = 2. Then G ∈ ℱ.
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Original Russian Text Copyright © 2009 Li Y., Qiao S., and Wang Y.
The authors were supported in part by the NSFC (No. 10871210), the NSF of Guangdong (No. 06023728), and the ARF (GDEI).
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Guangzhou. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 3, pp. 587–595, May–June, 2009.
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Li, Y., Qiao, S. & Wang, Y. A note on a result of Skiba. Sib Math J 50, 467–473 (2009). https://doi.org/10.1007/s11202-009-0052-1
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DOI: https://doi.org/10.1007/s11202-009-0052-1