Editing Abstract polytope (section)

== Exchange maps ==

Let ''Ψ'' be a flag of an abstract ''n''-polytope, and let −1&nbsp;<&nbsp;''i''&nbsp;<&nbsp;''n''. From the definition of an abstract polytope, it can be proven that there is a unique flag differing from ''Ψ'' by a rank ''i'' element, and the same otherwise. If we call this flag ''Ψ''<sup>(''i'')</sup>, then this defines a collection of maps on the polytopes flags, say ''φ''<sub>''i''</sub>. These maps are called '''exchange maps''', since they swap pairs of flags : (''Ψφ''<sub>''i''</sub>)''φ''<sub>''i''</sub>&nbsp;=&nbsp;''Ψ'' always. Some other properties of the exchange maps :
* ''φ''<sub>''i''</sub><sup>2</sup> is the identity map
* The ''φ''<sub>''i''</sub> generate a [[Group (mathematics)|group]]. (The action of this group on the flags of the polytope is an example of what is called the '''flag action''' of the group on the polytope)
* If |''i''&nbsp;−&nbsp;''j''| > 1, ''φ''<sub>''i''</sub>''φ''<sub>''j''</sub> = ''φ''<sub>''j''</sub>''φ''<sub>''i''</sub>
* If ''α'' is an automorphism of the polytope, then ''αφ''<sub>''i''</sub> = ''φ''<sub>''i''</sub>''α''
* If the polytope is regular, the group generated by the ''φ''<sub>''i''</sub> is isomorphic to the automorphism group, otherwise, it is strictly larger.
The exchange maps and the flag action in particular can be used to prove that ''any'' abstract polytope is a quotient of some regular polytope.