There are several characterizations of finite nilpotent groups (they are, as in wiki):

  1. $G$ is (finite) nilpotent group.

  2. Normalizer of every proper subgroup is bigger than the subgroup.

  3. Every maximal subgroup is normal.

  4. $G$ is direct product of Sylow subgroups.

The equivalence $ (1) \Longleftrightarrow (3)$ is often attributed to Wielandt, whereas, other equivalences are not attributed to anyone.

I wonder to see, who obtained these equivalent conditions. Can one give the historical reference to these equivalences?

Also, the theorem $$\mbox{the upper and lower central series of a group have same length}$$ gives a definition of nilpotent group with its nilpotency class. Philip Hall said in a paper that this remarkable theorem appears in the Speiser's book on Group Theory (~ 1909,very earlier book on the subject). Then, is this a theorem of Speiser, or it was known before him?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.