The conceptual background for this paper is given in part I by the same author [ibid., 1548-1556 (1993; see the preceding review Zbl 0786.20034)]. In this second part a method is presented for calculating the translation normalizers \(T_ N({\mathcal G})\) of Euclidean groups \(\mathcal G\) in \(E(n)\). The groups \(\mathcal G\) are given in terms of a point group \(G\subset O(n)\), a \(G\)-invariant translation group \(T_ G\subseteq V(n)\), the origin \(P\) and a system of non-primitive translations \(u_ G(G)\).
The group \(T_ N(G)\) only depends on the pair \((G,T_ G)\), so that one can write it as \(T_ N(G,T_ G)\). The normalizer \(T_ N(G,T_ G)\) consists of the translations \(\tau\) solution of the congruences of the shift functions: \(\varphi(g,\tau) = \tau-g\tau = 0\mod T_ G\). They are classified according to:
1. The character of the translation group \(T_ G\). One distinguishes between discrete translation subgroups \((T_ d)\) and continuous ones \((T_ c)\) of \(T_ G = T_ d\oplus T_ c\). Considered are also the subspaces of \(V(n)\) spanned on the reals by those subgroups \(V_ d = \langle T_ d\rangle_ R\) and \(V_ c = T_ c\) of dimension \(k_ d\) and \(k_ c\), respectively.
2. The reducibility (or the decomposability) of the point group represented by its action on \(T_ G\).
Considered are space groups (where \(T_ G = T_ d\) and \(k_ d = n\)), subperiodic groups with translation groups \(V_ c\) and \(k_ c<n\) (continuous case), \(T_ d + V_ c\), with \(k_ d + k_ c < n\) and \(k_ d,k_ c > 0\) (semicontinuous case), \(T_ d,k_ d < n\) (discrete case), respectively, and point groups \(G \subset O(n)\).
In a first step, the continuous part of the translation normalizer is determined. In a second step one considers the reducibility of the point group involved, as translation normalizers for reducible cases can be determined from the irreducible components. The third step depends on whether the point group \(G\) represented on the discrete part \(T_ d\) contains or not the trivial representation.
Examples of calculations of translation normalizers are given. In the Appendix a table lists these normalizers for discrete, continuous and semi-continuous Euclidean groups up to the dimension three.