, Volume 21, Issue 2, pp 289-298

All Polytopes Are Quotients, and Isomorphic Polytopes Are Quotients by Conjugate Subgroups

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In this paper it is shown that any (abstract) polytope \(\cal P\) is a quotient of a regular polytope \(\cal M\) by some subgroup N of the automorphism group W of \(\cal M\) , and also that isomorphic polytopes are quotients of \(\cal M\) by conjugate subgroups of W . This extends work published in 1980 by Kato, who proved these results for a restricted class of polytopes which he called ``regular''. The methods used here are more elementary, and treat the problem in a purely nongeometric manner.