科研之友
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摘要
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 boolean AND 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 U and let G be a group with E a normal subgroup of G such that G/E is an element of F. Suppose that each noncyclic Sylow p-subgroup P of F*(E) has a subgroup D such that 1 < vertical bar D vertical bar < vertical bar P vertical bar and all subgroups H of P with order vertical bar H vertical bar = vertical bar D vertical bar are weakly s-permutable in G for all p is an element of pi(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 vertical bar D vertical bar = 2. Then G is an element of F.
