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Abstract  

A characterisation of finite soluble groups in which Sylow permutability is a transitive relation by means of subgroup embedding properties enjoyed by all the subgroups is proved. The key point is an extension of a subnormality criterion due to Wielandt.

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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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115 – 187 . [2] Beidleman , James C. , Brewster , Ben , Robinson , Derek J. S. 1999 Criteria for permutability to be transitive in finite groups J. Algebra 222 400 – 412 10.1006/jabr

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. Algebra , 371 ( 2012 ), 250 – 261 . [9] S chmid , P. , Subgroups permutable with all Sylow subgroups , J. Algebra , 207 ( 1998 ), 285 – 293

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