On pyramidal groups of prime power degree

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 $\left|C\right|=m$. We call groups with this property m-pyramidal. We prove that, if m is an odd prime power $p^k$, with $p\ne 7$, 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.