Isometric groups of non-rigid molecules: homomorphisms and reformulation of a theorem on semi-direct product factorization

Abstract

The structure of isometric groups of semi-rigid molecules is considered from the point of view of homomorphisms relating various representations defining isometric groups. From the kernels of the group homomorphisms the existence of a number of invariant subgroups (, , etc.) is demonstrated.

One of these homomorphisms is shown to relate the problem of the decomposition of isometric groups into semi-direct products to the analogous problem of molecular covering groups. Based on this relation a theorem reported earlier is reformulated by stating explicitly general conditions under which proper semi-factorization is possible or impossible respectively. The latter situation exists only for certain normal cyclic subgroups of cyclic and dihedral groups, whose order is a product of powers of primes.