Symmetrizers are defined as such symmetry operations that contribute to enhancing the symmetry of the point group
GX of a crystal to that of
GD of its X-ray diffraction pattern when
GX<
GD. In order to look for possible types of symmetrizer, the crystal structure under consideration is assumed to consist of substructures identical to each other and distributed according to parallel shifts from one to the others. Such a structure is aptly described by a space groupoid, the substructures being the geometric representation of its kernel. It has been known that if the structure gives rise to an enhanced diffraction (vector) symmetry, the symmetry of the substructure is equal to the enhanced symmetry, and the conditions for the hull of the space groupoid to satisfy when the diffraction symmetry is enhanced are sought in the present study. From the consideration of the symmetry of the vector set in general, the relation of direct sum is derived. This relation is then applied to the hull of the present space groupoid, and it is concluded that if the relation of direct sum holds in the hull and if the vector subsets, each produced by a term in the direct sum, are symmetrically related to each other by an operation
g, such a structure may be constructed as causing an enhanced diffraction symmetry, for which
g is a symmetrizer.
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