LAYER HOMOLOGY GROUPS. A.F.Palistrant, A.A.Zadorozhny, Geometry Department of Moldova State University, Kishinev

Crystal homology by V.I.Miheev, affinely adequate to usual symmetry, but extending traditional classification of various crystal forms, is used to generalize layer symmetry groups.

Essence of notion of crystal homology is as follows. Let F be a finite figure, and S be its discrete symmetry group. Group H of affine transformations, obtained from symmetry group S of the figure mentioned, by means of some affine transformation [[sigma]] of this figure according to the law H=[[sigma]]S[[sigma]]^-1 is called the homology group of the figure considered, and any h=[[sigma]]s [[sigma]]^-1 from H, where s is from S, is called the transformation of homology of the given figure. Transformations of homology of finite figure are equiaffine and transfer symmetry group S of this figure into its homology group H, and elements of symmetry of group S into elements of homology of group H.

The apperence of tablet homology groups, preserving a plane and a point on it in three-dimensional space, as well as the line crossing this plane in the mentioned point, gives the possibility to solve the task of generalizing layer symmetry groups, preserving in the same space the only plane and no point, and no line on it, up to the corresponding homology groups. The reason is that, according to the general theory, to obtain the groups we are interested in, we should use (according to the above mentioned law) such affine transformation [[sigma]] for each layer symmetry group S, that the point subgroup of "rotations", being a part of group S, is transformed into one of the 145 tablet homology groups obtained earlier.

By the above mentioned method 414 layer homology groups were obtained from 80 layer symmetry groups.