Algorithms for Nilpotent Linear Groups

Abstract

In this thesis, we develop practical algorithms for irreducibility and primitivity testing of finite nilpotent linear groups defined over various fields of characteristic zero, including number fields and rational function fields over number fields. For a reducible group, we can construct a proper submodule. For an irreducible but imprimitive group, we exhibit a system of imprimitivity. An implementation of the above-mentioned algorithms is publicly available as a stand-alone package and also included in recent versions of the Magma computer algebra system.

In addition to the above, we also develop an algorithm for deciding irreducibility of possibly infinite nilpotent linear groups defined over number fields. Finally, we study the structure of primitive finite nilpotent linear groups over number fields.