On pyramidal groups whose number of involutions is a prime power
Xiaofang Gao, Martino Garonzi
TL;DR
This work analyzes the group-theoretic underpinnings of m-pyramidal Kirkman triple systems, establishing when all such groups are solvable for odd prime powers m = p^k with p ≠ 7. The authors develop a five-step prime-case framework using centralizers, Frattini theory, and primitive/almost-simple group structure, and they show that solvability is equivalent to k being odd or the exceptional case m = 9. They also classify the possible orders of m-pyramidal groups when m is prime, revealing a precise arithmetic description X_m = Y_m ∪ Z_m tied to Mersenne primes and the factorization of m−1. The results connect design theory with deep aspects of permutation groups, including 2-transitive affine actions and minimal almost-simple groups, and they provide explicit nonsolvable constructions in the even-k, non-9 prime-power regime.
Abstract
A Kirkman Triple System $Γ$ is called $m$-pyramidal if there exists a subgroup $G$ of the automorphism group of $Γ$ that fixes $m$ points and acts regularly on the other points. Such group $G$ admits a unique conjugacy class $C$ of involutions (elements of order $2$) and $|C|=m$. We call groups with this property $m$-pyramidal. We prove that, if $m$ is an odd prime power $p^k$, with $p \neq 7$, then every $m$-pyramidal group is solvable if and only if either $m=9$ or $k$ is odd. The primitive permutation groups play an important role in the proof. We also determine the orders of the $m$-pyramidal groups when $m$ is a prime number.