Groups, isomorphic

Mapping L on another space L1, using the operator S creates an isomorphic group of operators in the space L which are the transforms of A, B,... by the operator S ... 

Carbonate cement content in the studied rocks varies from 0 to 45 vol. %, mostly forming the pore filling. The following carbonates were observed (Koztowska 2004) siderite, Fe-dolomite, ankerite and Fe-calcite. The term siderite corresponds to minerals from the isomorphic group FeCOs-MgCOs with 60-100 mol percent FeCOs. Most siderites fall into the interval siderite - sideroplesite (Fig. 3). 

The rhombic bipyramidal crystals of potassium sulphate form an isomorphous group with those of the corresponding salts of rubidium and csesium. The axial ratios and mol. vol. are indicated in Table XXXVI the axial ratios and topical parame.ters of the three last-named salts were determined by A. E. H. Tutton.26... 

Paraphrasing this definition, we may say that isomorphic groups have group tables of the same structure or form although they may differ in respect of the notation and nature of their elements and differ in... 

Isobutane, dipole moment of, 225 Isomorphic groups, 392 Isotopes, table of properties of, 470 Isotopic substitution in ESR, 379 in IR spectroscopy, 244 in microwave spectroscopy, 221-223 in NMR, 354-355... 

If multiplication is preserved in the mapping of G on to G, eq. (7), then any properties of G, G that depend only on the multiplication of group elements will be the same in G, G. Thus isomorphous groups have the same multiplication table and class structure. 

Table 1. Characters for the rotation group Cg and the isomorphic groups Sg aond Cij, which are hemihedrics of the holohedric Cg/,. Sg is the holohedtic triclinic group...

The 3-r symbols of the group 0 can be taken as 3-F symbols also for the isomorphic group Ta, and these symbols may again be used for the tetrahedral rotation group T when a real basis is sufficient. 

Character tables for some isomorphous groups and their double groups... 

When the dements of two groups can be mapped bijectivdy onto each other, such that the binary relations expressed in their multiplication tables are maintained, the two groups are said to be isomorphic. As a consequence, isomorphic groups have identical character tables. Both... 

The following isomorphous groups have been established, and the existence of more is highly probable —... 

Indeed, the only plausible explanation of the existence of isomorphous groups of compounds is, that the elements characterising those groups are isomorphous, and that hence their analogous compounds are so. If we assume that arsenic and phosphorus are isomorphous, then we see that As, 0, must be isomorphous with P, 0, since the oxygen in both is, of course, the same. In like manner, arseniate of soda. As, O, 2 Na 0, HO, 24 aq., must be isomorphous with phosphate of soda, P, 0, 2 Na O, HO, 24 aq., since all the elements in these two salts are the same in nature, number, and arrangement, except that As, in the first is replaced by P, in the second, and the elements As and P have bwn assumed to possess the same form. 

The first two isomorphous groups listed in Table XXVIII may be noted to have one complex common to both Ni(OEP). Members of the two series differ in the extent of their n-n interactions members of the first series were catalogued into Group W and members of the second series into Group I. The obvious question of why Ni(OEP) crystallizes in both series is not clear, however, preliminary evidence suggests that the occurence of the compound in the first series results from a template effect. In our hands the first crystalline form of Ni(OEP) is found to always contain small amounts of HjOEP the second contains less or none. Further work is needed to completely clarify the matter. The crystal packing effects of the third series were mentioned earlier and are detailed elsewhere ... 

Polyvinyl alcohol (a) and polyvinyl fluoride (b) have isomorphous groups so can crystallize even if they are atactic... 

SECS also uses the stereochemical graph isomorphism group within a transform to correctly determine if atoms or groups are chemically equivalent "IF ATOM 1 AND ATOM 2 ARE EQUIVALENT THEN..." Complete details of the symmetry recognition algorithm and its application are given elsewhere. ... 

Isomorphic groups, 91, 320 Isotope effect, vibrational, influence of anharmonicity, 183 on intensities, 191. potential energy invariance, 171, 182, 192... 

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