The multiplication table - an example

Thus C3 and crd do not commute. These operator equalities in eqs. (l)-(4a) are true for any initial configuration. For example, 

Exercise 2.2-2 Find the products Cgae and reC3. The multiplication table for this set of operators G E Cf C3 rrd ae at is shown in Table 2.3. The complete multiplication 

Any set with the four properties (a)-(d) forms a group therefore the set G is a group for which the group elements are point symmetry operators. This point group is called C3v or 3m, because the pyramid has these symmetry elements a three-fold principal axis and a vertical mirror plane. (If there is one vertical plane then there must be three, because of the three-fold symmetry axis.) 

Exercise 2.2-4 Are the groups C3v and S(3) isomorphous [Hint. Compare Table 2.3 with Table 1.3.] 

Exercise 2.2-1 The orientation of the triangular base of the pyramid is shown for each of the indistinguishable configurations. 

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