Finite groups

28.2 FINITE GROUPS
28.2 Finite groups
Whilst 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 under
ordinary multiplication, restricted sets can do so if the operation involved is multiplication (mod N) for suitable values of N; this operation will be explained below.
As a simple example of a group with only four members, consider the set S
defined 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, since Y × Z = Z × Y , the full set
of different products is
1 × 1 = 1,
3 × 3 = 1,
5 × 5 = 1,
7 × 7 = 1.
1 × 3 = 3,
3 × 5 = 7,
5 × 7 = 3,
1 × 5 = 5,
3 × 7 = 5,
1 × 7 = 7,
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 = Y for all members Y of the set. Further, for each element Y
of the set there is an element Z (equal to Y , 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 set S is an Abelian group
of order 4.
It is convenient to present the results of combining any two elements of a
group 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 written X • Y then X is taken from the left-hand
column and Y is 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 table
to present the rows and columns with their headings in random orders, it is
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GROUP THEORY
1
3
5
7
1
1
3
5
7
3
3
1
7
5
5
5
7
1
3
7
7
5
3
1
Table 28.1 The table of products for the elements of the group S = {1, 3, 5, 7}
under multiplication (mod 8).
usual to list the elements in the same order in both the vertical and horizontal
headings in any one table. The actual order of the elements in the common list,
whilst arbitrary, is normally chosen to make the table have as much symmetry as
possible. This is initially a matter of convenience, but, as we shall see later, some
of the more subtle properties of groups are revealed by putting next to each other
elements of the group that are alike in certain ways.
Some simple general properties of group multiplication tables can be deduced
immediately from the fact that each row or column constitutes the elements of
the group.
(i) Each element appears once and only once in each row or column of the
table; this must be so since G • X = G (the permutation law) holds.
(ii) The inverse of any element Y can be found by looking along the row
in which Y appears in the left-hand column (the Y th row), and noting
the element Z at the head of the column (the Zth column) in which
the identity appears as the table entry. An immediate corollary is that
whenever the identity appears on the leading diagonal, it indicates that
the corresponding header element is of order 2 (unless it happens to be
the identity itself).
(iii) For any Abelian group the multiplication table is symmetric about the
leading diagonal.
To get used to the ideas involved in using group multiplication tables, we now
consider two more sets of integers under multiplication (mod N):
S = {1, 5, 7, 11}
S = {1, 2, 3, 4}
under multiplication (mod 24), and
under multiplication (mod 5).
These have group multiplication tables 28.2(a) and (b) respectively, as the reader
should verify.
If tables 28.1 and 28.2(a) for the groups S and S are compared, it will be seen
that they have essentially the same structure, i.e if the elements are written as
{I, A, B, C} in both cases, then the two tables are each equivalent to table 28.3.
For S, I = 1, A = 3, B = 5, C = 7 and the law of combination is multiplication
(mod 8), whilst for S , I = 1, A = 5, B = 7, C = 11 and the law of combination
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28.2 FINITE GROUPS
(a)
1
5
7
11
1
1
5
7
11
5
5
1
11
7
7
7
11
1
5
11
11
7
5
1
(b)
1
2
3
4
1
1
2
3
4
2
2
4
1
3
3
3
1
4
2
4
4
3
2
1
Table 28.2 (a) The multiplication table for the group S = {1, 5, 7, 11} under
multiplication (mod 24). (b) The multiplication table for the group S =
{1, 2, 3, 4} under multiplication (mod 5).
I
A
B
C
I
I
A
B
C
A
A
I
C
B
B
B
C
I
A
C
C
B
A
I
Table 28.3 The common structure exemplified by tables 28.1 and 28.2(a).
1
i
−1
−i
1
1
i
−1
−i
i
i
−1
−i
1
−1
−1
−i
1
i
−i
−i
1
i
−1
Table 28.4 The group table for the set {1, i, −1, −i} under ordinary multiplication of complex numbers.
is multiplication (mod 24). However, the really important point is that the two
groups S and S have equivalent group multiplication tables – they are said to
be isomorphic, a matter to which we will return more formally in section 28.5.
Determine the behaviour of the set of four elements
{1, i, −1, −i}
under the ordinary multiplication of complex numbers. Show that they form a group and
determine whether the group is isomorphic to either of the groups S (itself isomorphic to
S ) and S defined above.
That the elements form a group under the associative operation of complex multiplication
is immediate; there is an identity (1), each possible product generates a member of the set
and each element has an inverse (1, −i, −1, i, respectively). The group table has the form
shown in table 28.4.
We now ask whether this table can be made to look like table 28.3, which is the
standardised form of the tables for S and S . Since the identity element of the group
(1) will have to be represented by I, and ‘1’ only appears on the leading diagonal twice
whereas I appears on the leading diagonal four times in table 28.3, it is clear that no
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