Plakosh A. Application of matrix problems in group theory

The thesis is devoted to the study of matrix problems and the calculation of cohomology related to the problem of classification of Chernikov $p$-groups. The Chernikov $p$-groups with an elementary Abelian top and the basis of rank $2$ are described. A classification of pairs of skew-symmetric forms over a field is given up to the weak congruence. A relation was found between the integral representations of the Kleinian $4$-group and the representations of a quiver of type $\tilde{D}_4.$ The classification of integer representations of the Kleinian $4$-group is given up to the weak equivalence. A new resolution of the trivial lattice over a finite Abelian group is constructed, which is better suited for the calculation of cohomology, and its connection with the standard resolution is established. New duality relations for lattices over finite groups are established. The cohomology of irreducible lattices and their dual modules over finite Abelian groups is calculated. The cohomology of modules dual to inexact lattices over the Kleinian $4$-group is calculated.