Highest rank of a polytope for An

Abstract

We prove that the highest rank of a string C-group constructed from an alternating group An is 3 if n = 5; 4 if n = 9; 5 if n = 10; 6 if n = 11; and the floor of of (n-1)/2 if n>=12. Moreover, if n = 3; 4; 6; 7 or 8, the group An is not a string C-group. This solves a conjecture made by the last three authors in 2012.