7 problems

A CPR graph is a permutation representation graph associated with a string C-group representation. For each $n$, consider the permutation representation graphs of rank $r\geq n/2$…

Interval-rank conjecture. The group $A_{11}$ is the only finite simple group whose set of ranks of string C-group representations is not an interval in the set of integers.

Rank-three conjecture for $PGL(2,q)$. The maximum rank of a string C-group representation of $PGL(2,q)$ should be three except when $q=5$.

Constant-count conjecture. The number of string C-group representations of rank $n-i$ for $S_n$ with $1\leq i\leq (n-3)/2$ is a constant independent of $n$.

Rank-interval conjecture. The group $\operatorname{Alt}(11)$ is the only finite simple group whose set of ranks of string C-group representations is not an interval in the set of i…

A string C-group of alternating type is a string C-group whose underlying group is an alternating group; for $n\geq 12$, the known constructions give alternating groups a rank at l…

Let $n\geq 12$. An abstract regular polytope with automorphism group $\mathrm{A}_n$ has rank $d$, where rank means the number of involutory generators in its string C-group represe…