28.2 FINITE GROUPS
28.2 Finite groupsWhilst many properties of physical systems (e.g. angular momentum) are related
to the properties of infinite, and, in particular, continuous groups, the symmetry
properties of crystals and molecules are more intimately connected with those of
finite groups. We therefore concentrate in this section on finite sets of objects that
can be combined in a way satisfying the group postulates.
Although it is clear that the set of all integers does not form a group underordinary multiplication, restricted sets can do so if the operation involved is multi-
plication (modN) for suitable values ofN; this operation will be explained below.
As a simple example of a group with only four members, consider the setSdefined as follows:
S={ 1 , 3 , 5 , 7 } under multiplication (mod 8).To find the product (mod 8) of any two elements, we multiply them together in
the ordinary way, and then divide the answer by 8, treating the remainder after
doing so as the product of the two elements. For example, 5×7 = 35, which on
dividing by 8 gives a remainder of 3. Clearly, sinceY×Z=Z×Y, the full set
of different products is
1 ×1=1, 1 ×3=3, 1 ×5=5, 1 ×7=7,
3 ×3=1, 3 ×5=7, 3 ×7=5,
5 ×5=1, 5 ×7=3,
7 ×7=1.The first thing to notice is that each multiplication produces a member of the
original set, i.e. the set is closed. Obviously the element 1 takes the role of the
identity, i.e. 1×Y=Yfor all membersYof the set. Further, for each elementY
of the set there is an elementZ(equal toY, as it happens, in this case) such that
Y×Z= 1, i.e. each element has an inverse. These observations, together with the
associativity of multiplication (mod 8), show that the setSis an Abelian group
of order 4.
It is convenient to present the results of combining any two elements of agroup in the form of multiplication tables – akin to those which used to appear in
elementary arithmetic books before electronic calculators were invented! Written
in this much more compact form the above example is expressed by table 28.1.
Although the order of the two elements being combined does not matter here
because the group is Abelian, we adopt the convention that if the product in a
general multiplication table is writtenX•YthenXis taken from the left-hand
column andYis taken from the top row. Thus the bold ‘ 7 ’ in the table is the
result of 3×5, rather than of 5×3.
Whilst it would make no difference to the basic information content in a tableto present the rows and columns with their headings in random orders, it is