# A result from Peter McMullen's thesis

The classical definition of regular polytopes is recursive. It says that a polytope is regular if its facets and vertex figures (both smaller-dimensional polytopes) are regular.

The modern definition goes as follows. Let $P\subseteq\mathbb{R}^n$ be a polytope centered at the origin and let $\mathrm{Aut}(P)\leq O(n)$ be its automorphism group. We say that $P$ is regular if its automorphism group acts transitively on maximal flags of faces.

However, there are many equivalent definitions of regularity. Let's say that a polytope is $d$-regular if its automorphism group is transitive on $d$-dimensional faces. The following theorem is stated in several places (for example in McMullen and Schulte's "Abstract Regular Polytopes", pages 9-10):

Theorem: Let $P$ be an $n$-dimensional polytope. If $P$ is $d$-regular for all $0\leq d\leq n-1$ then $P$ is regular.

All statements of this theorem I've seen refer to Peter McMullen's 1968 thesis from the University of Birmingham, which I don't have access to.

So here's my question: Does anyone know where I can find a proof of this theorem or how to gain access to Peter McMullen's thesis?

Bonus Problem: How dependent/independent are the notions of $d$-regularity for different $d$?

Thanks.

-
This seems to need a clarification on automorphisms: clearly the cyclic group of order $n$ acting naturally on the regular $n$-gon is transitive on its vertices and edges, but not on flags. So the statement seems to boil down to a finite Coxeter group in the background... – Dima Pasechnik Feb 20 at 23:23