Regular maps with nilpotent automorphism groups

We study regular maps with nilpotent automorphism groups in detail. We prove that every nilpotent regular map decomposes into a direct product of maps H × K , where Aut ( H ) is a 2 -group and K is a map with a single vertex and an odd number of semiedges. Many important properties of nilpotent maps follow from this canonical decomposition, including restrictions on the valency, covalency, and the number of edges. We also show that, apart from two well-defined classes of maps on at most two vertices and their duals, every nilpotent regular map has both its valency and covalency divisible by 4. Finally, we give a complete classification of nilpotent regular maps of nilpotency class 2.