On pyramidal groups of prime power degree

A Kirkman Triple System Gamma is called m- pyramidal if there exists a subgroup G of the automorphism group of Gamma 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 not equal 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. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.