# An Alternate Proof to Derek Robinson’s 1968 Local Characterization Theorem on *T*-Groups

## Abstract

Define \(\mathcal {T}\) to be the class of groups in which normality is a transitive relation. Define \({\mathscr {C}}_p\) to be the class of finite groups *G* for which each subgroup of a Sylow *p*-subgroup of *G* is normal in the corresponding Sylow normalizer. In [5] Robinson proved that a finite group satisfies \({\mathscr {C}}_p\) for all primes *p* if and only if it is a finite solvable \(\mathcal {T}\)-group. Here a new proof of this classic and influential result is presented.

## Keywords

Solvable Group T-Group Supersolvable Group Sylow Subgroup## Notes

### Acknowledgements

Derek Robinson’s 1968 paper [5] on *T*-groups is the first mathematical article that the author read as a Ph.D. student under Professor James (Jim) Clark Beidleman’s guidance. The author fondly remembers working through details of the paper and meeting regularly with Dr. Beidleman who used the result in his work often. He once suggested that the author’s alternate proof presented here (arrived at many years after working through the original proof) be published. Sadly, Dr. Beidleman passed away recently [4]; this article is dedicated to his memory.

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