# Characterizations of Nilpotent Groups

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 Can one give the historical reference to these equivalences?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?