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] Asaad , M. and Heliel , A. A. , On S-quasinormality embedded subgroups of finite groups , J. Pure Appl. Algebra , 165 ( 2001 ), 129 – 135 . [2] Ballester

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A saad , M. , Products of finite groups , De Gruyter Expositions in mathematics , vol. 53 . Walter de Gruyter, Berlin ( 2010 ). [2] B uckley , J. , Finite

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De Caen, D., Gregory, D. A., Hughes, I. G. and Kreher, D. L. , Near-factors of finite groups, Ars Combinatoria , 29 (1990), 53–63. MR 91c :20037 Kreher D. L

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Abstract  

Let G be a finite group. A PT-group is a group G whose subnormal subgroups are all permutable in G. A PST-group is a group G whose subnormal subgroups are all S-permutable in G. We say that G is a PTo-group (respectively, a PSTo-group) if its Frattini quotient group G/Φ(G) is a PT-group (respectively, a PST-group). In this paper, we determine the structure of minimal non-PTo-groups and minimal non-PSTo-groups.

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Agrawal, R. K. , Generalized center and hypercenter of a finite group, Proc. Amer. Math. Soc. , 54 (1976), 13–21. Agrawal R. K

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References [1] Asaad , M. 1988 On the solvability of finite groups Arch. Math. 51 289 – 293

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References [1] Ballester-Bolinches , A. , Ezquerro , L. M. 2006 Classes of Finite Groups Springer Dordrecht . [2

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References [1] A saad , M. , H eliel , A. A. and A l -S homranim , A. A. , On weakly ℌ -subgroups of finite groups , Comm. Algebra , 40 ( 2012 ), 3540

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Abstract  

We use ?-quasinormal condition on minimal subgroups to Characterize the structure of a finite group through the theory of formation. We give some equivalent conditions of a nilpotent group or a saturated formation containing the nilpotent groups. Our results generalize earlier theorems of Yokoyama, Ballester-Bolinches and Pedraza Aguilera.

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Abstract  

Let G be a finite group. For a finite p-group P the subgroup generated by all elements of order p is denoted by Ω1(p). Zhang [5] proved that if P is a Sylow p-subgroup of G, Ω1(P) ≦ Z(P) and N G(Z(P)) has a normal p-complement, then G has a normal p-complement. The object of this paper is to generalize this result.

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