Group theory

28.1 GROUPSLMKFigure 28.2 Reflections in the three perpendicular bisectors of the sides ofan equilateral triangle take the triangle into itself.(iii) Forming the product of each element of G with a fixed element X of Gsimply permutes the elements of G; this is often written symbolically asG • X = G. If this were not so, and X • Y and X • Z were not differenteven though Y and Z were, application of the cancellation law would leadto a contradiction. This result is called the permutation law.In any finite group of order g, any element X when combined with itself toform successively X 2 = X • X, X 3 = X • X 2 , . . . will, after at most g − 1 suchcombinations, produce the group identity I. Of course X 2 , X 3 , . . . are some ofthe original elements of the group, and not new ones. If the actual number ofcombinations needed is m−1, i.e. X m = I, then m is called the order of the elementX in G. The order of the identity of a group is always 1, and that of any otherelement of a group that is its own inverse is always 2.Determine the order of the group of (two-dimensional) rotations and reflections that takea plane equilateral triangle into itself and the order of each of the elements. The group isusually known as 3m (to physicists and crystallographers) or C 3v (to chemists).There are two (clockwise) rotations, by 2π/3 and 4π/3, about an axis perpendicular tothe plane of the triangle. In addition, reflections in the perpendicular bisectors of the threesides (see figure 28.2) have the defining property. To these must be added the identityoperation. Thus in total there are six distinct operations and so g = 6 for this group.To reproduce the identity operation either of the rotations has to be applied three times,whilst any of the reflections has to be applied just twice in order to recover the originalsituation. Thus each rotation element of the group has order 3, and each reflection elementhas order 2. A so-called cyclic group is one for which all members of the group can begenerated from just one element X (say). Thus a cyclic group of order g can bewritten asG = I, X, X 2 , X 3 , . . . , X g−1 .1047