Imprimitive Complex Reflection Groups

As a class of finite reflection groups and different from finite real reflection groups, finite complex reflection groups are studied by more and more people in recent years. Among finite complex reflection groups, the imprimitive complex reflection groups seem to attract more attention of people because of their good properties in combinatorics. Recently, Shi has done much work on the presentations of imprimitive complex reflection groups(see [26],[27], [28]). Based on Shi's work, we continue to study the following several problems:1. Automorphism group of the imprimitive complex reflection group G(m,p, n) Shi defined a congruence relation on the set of all the presentations of the imprimitive complex reflection group G(m, p, n) (two presentations are congruent if and only if their is a congruence map between them)(see [27],[28]). We define a strongly congruence relation on the set of all the presentations of the imprimitive complex reflection group G(m, p, n) (two presentations are strongly congruent if and only if their is a strongly congruence map between them), such that any strongly congruence map between two presentations of G(m, p, n) can be extened to be an automorphism of the group G(m, p, n), and any two presentations of G(m, p, n) which are strongly congruent are definitely congruent but not the converse. By applying Shi's results on congruent presentations, we can describe the automorphism group Aut(m, p, n) of the imprimitive complex reflection group G(m, p, n), which consists of all the automorphisms each of whom sends a reflection to a reflection.2. Some properties of the group Aut(m, p, n)After we have determined the automorphism group Aut(m, p, n) of the imprimitive complex reflection group G(m, p, n), we make some more study of its properties, which contains the research of the order of Aut(m, p, n), the structure of Aut(m, p, n), the subgroups of Aut(m, p, n) and the center of Aut(m, p, n).3. The reflection subgroups and the subsystems of the root system of the imprimitive complex reflection group G(m, p, n)starting from the root graph of a complex reflection group, Hughes defined an extended Cohen diagram, and gave an algorithm to compute all the subsystems of the root system of that group(see [15], [16]). But for a complex reflection group without a root...