String C-group representations of alternating groups

TL;DR

This paper demonstrates that for all sufficiently large alternating groups, there exist string C-group representations of all ranks within a certain range, except for A11, which is unique in this aspect.

Contribution

It establishes the existence of string C-group representations of various ranks for all alternating groups except A11, expanding understanding of their symmetry properties.

Findings

A_n has string C-group representations of all ranks in a specified interval for n ≥ 12.

The only exception among alternating groups is A11, which does not have a continuous set of ranks.

Provides a comprehensive classification of ranks for alternating groups' string C-group representations.

Abstract

We prove that for any integer $n\geq 12$, and for every $r$ in the interval $[3, \ldots, \lfloor (n-1)/2\rfloor]$, the group $A_n$ has a string C-group representation of rank $r$ therefore showing that the only alternating group whose set of ranks is not an interval is $A_{11}$.