Rearrangement theorem

Inspection of Table 3-4.1 shows that each row or column contains each element once and once only. This is true for all group tables and the proof, the Rearrangement Theorem, is given in Appendix A.3-1. 

If SR — T, thenD( )Z)(R) = Z)(T), see Appendix A.5-5, and since the primed matrices are obtained through a similarity transformation, they are equivalent to the unprimed ones and we have D (S)iy(R) = D (T). Furthermore, by the Rearrangement Theorem (Appendix A.31), as R runs over the symmetry operations of the point group, so dees SR( T). Hence... 

The penultimate step is true since as S runs through the operations of the group so, by the Rearrangement Theorem (Appendix A.31), does RS(= T). Replacing T by S gives the final stop. 

By the Rearrangement Theorem (Appendix A.3-1) there will only be one value ofj for which R Rf Ra id only one value ofj for which Hr1 , = Rb. The sum over j in eqn (A.7-2.2) will vanish unless for some single j value both D%g(Ra) and are simultaneously non-zero and this will occur,... 

We now prove an important theorem about group multiplication tables, called the rearrangement theorem. 

Using the Rearrangement Theorem to replace G with GU gives... 

Because gk is a unique element of g,, if each element of g, is multiplied from the left, or from the right, by a particular element g, of g, then the set g, is regenerated with the elements (in general) re-ordered. This result is called the rearrangement theorem... 

Note that g, means a set of elements of which g,- is a typical member, but in no particular order. The easiest way of keeping a record of the binary products of the elements of a group is to set up a multiplication table in which the entry at the intersection of the g,th row and gyth column is the binary product g - g - = gk, as in Table 1.1. It follows from the rearrangement theorem that each row and each column of the multiplication table contains each element of G once and once only. 

There are only two groups of order 4 that are not isomorphous and so have different multiplication tables. Derive the multiplication tables of these two groups, G4 and G4. [Hints. First derive the multiplication table of the cyclic group of order 4. Call this group G4. How many elements of G4 are equal to their inverse Now try to construct further groups in which a different number of elements are equal to their own inverse. Observe the rearrangement theorem.]... 

Because binary composition is unique (rearrangement theorem) the same restriction of only one non-zero element applies to the rows of T8. 

The reader will note that coefficients have been introduced into Table IV and elsewhere as needed. According to the rearrangement theorem, Cotton (33, p. 6), Each row and each column in the group addition table lists each of the group elements once and only once. From this, it follows that no two rows can be identical nor can any two columns be identical. Thus, each row and each column is a rearranged list of the... 

Some properties of groups 3-6. Classification of point groups 3-7. Determination of molecular point groups A. 3-1. The Rearrangement Theorem Problems... 

By the Rearrangement Theorem (Appendix A.3-1) there will only be one value ofj for which Ri /ij = and only one value ofj for which... 

Solution. The structure of BH3 is shown below, where the labels on the H atoms are imaginary. VSEPR theory predicts a trigonal planar electron and molecular geometry with bond angles of 120. The complete set of symmetry operations is , C3, 3, C2, C2, C2", tTh, S3, 83, c7v, CTv2> w3- The multiplication table is shown below. Once about half of the products were determined, the rearrangement theorem was used to help complete the multiplication table. 

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