TITLE: Why is a group of order 135 nilpotent?
QUESTION [6 upvotes]: Why does order 135 imply nilpotent?

REPLY [3 votes]: Every group of order $135$ is nilpotent because $135$ is a nilpotent number.  This  note explains what that means.  (It gives a simple arithmetic criterion on $n$ that is necessary and sufficient for every group of order $n$ to be nilpotent.)  
Let me be upfront about the fact that the linked to paper, while short and nicely written, is not fully self-contained: at a key point it appeals to some results of Schmidt (that you could read about in a "serious" group theory text -- e.g. Robinson's GTM on group theory -- but not in the standard introductory algebra texts, so far as I know).  So this answer is intended for "enrichment".