Summary of Results

All colour groups with same group of permutations (A,A')_n forms a ``chromomorphic'' class. It is important to note that symmetry properties of the phase transition are determined from the class (A,A')_n. The results of analysis made in the frame (A,A')_n is valid for all the space groups G from class (A,A')_n. Hence, from mathematical point of view, it is enough to analyze all possible transitions $A \longrightarrow A'$ for the finite groups A. All phase transitions without change of translational symmetry are described with colour groups which translational subgroup preserves all colours. Total number of these groups is 2571[6].

In case of change of the translational symmetry, the number of colour groups increases. The main object of the diploma thesis was investigation in this more complicated case. We restrict our research to colour groups describing transitions to commensurate phases i.e. with irreducible representations which wave vector is from the surface of Brillouin zone. We use images of the irreps from book [9].

The original results from the diploma thesis can be summarized as follows:

1.

Universal pack of programs for generating and investigating in the field of matrix and permutational groups is developed. All programs are written in C language and can be compiled and used under Unix environmental.

2.

All 115 abstract groups A, isomorphic to the matrix groups-images are generated in an explicit form and the corresponding multiplication tables are produced, all classes of conjugated elements and class-multiplication coefficients are tabulated.

3.

Full set of subgroups for each image A is found. All classes of conjugated subgroups and subgroups with $\mbox{ \rm Core }A' = C_1$ are listed.

4.

Derived groups and subgroups are classified in chromomorphic classes (A,A')_n.

5.

Transitive permutational representations D_A^A' for each chromomorphic class (A,A')_n are calculated.

6.

All isotropic subgroups of the faithful representations are found.

7.

All irreducible representations, of the crystallographic space groups, which have wave vector $\vec k$ from the surface of the Brillouin zone are classified on the base of images A.

8.

All results from the diploma thesis are saved on a CD-ROM in easy to use form.

9.

A set of phase transitions with 16-component order parameters in M₃B₇O₁3X is analyzed (for illustration). Found is full correspondence with other authors.

Work was supported in part by two grants (# F-468/95 of the Bulgarian National Sci. Foundation and #SU-125/95 of the Research Foundation of the University of Sofia). The results of this study are preparing for publication.