# The role of normalizers in the theory of crystallographic groups Acta Crystallographica Section A Foundations of Crystallography. Published: 1981-08-16. 0 citations. ## Authors - W. Fischer (Institute of Mineralogy): h-index 9; 205 citations; corresponding author ## Topics - X-ray Diffraction in Crystallography - Chemical Thermodynamics and Molecular Structure --- # 20. SYMMETRY AND ITS GENERALIZATION C-349 20.X-Ol ## Abstract After brief historical introduction the various definitions of color symmetry vrill be revie1•red:J compared and analyzed. Finally algoritlo~s for classification of color symmetries will be discussed and the role of the norwslizer concept will be exhibited. Let G be a group and H a supergroup of G. Then the largest subgroup of H that contains G as a normal subgroup is called the normalizer of G in H, NH(G). It therefore consists of all elements h of H that map G onto itself by conjugation: h-1 Gh=G. The subset CH(G) of NH(G) containing all elements h of H with h-1 gh=g for all elements g of G forms a normal subgroup of NH(G) called the centra- lizer of G in H. CH(G) is a supergroup of G if and only if G is Abelian. More specifically, let G be a