On Integral Groups . Ill : Normalizers

Methods for determining a generating set for the normalizer of a finite group of n X n integral matrices, i.e., an «-dimensional crystallographic point group, are discussed. Necessary and sufficient conditions for the finiteness of such a normalizer are derived, and several examples of the application of the methods to cases when the normalizer is infinite are presented. This is the third in a series of papers dealing with the finite subgroups of GL(n, Z). In the first two papers [3], [4], we discussed the integral classification of these groups, and, in this paper, we consider their normalizers in GL(«, Z), which are needed, e.g., for the determination of the «-dimensional space groups [2], [17]. As in the previous two papers, the discussion results in a complete determination for the case n = A. We refer to the first paper for basic definitions and notation.