Groups

28
Group theory
For systems that have some degree of symmetry, full exploitation of that symmetry
is desirable. Significant physical results can sometimes be deduced simply by a
study of the symmetry properties of the system under investigation. Consequently
it becomes important, for such a system, to identify all those operations (rotations,
reflections, inversions) that carry the system into a physically indistinguishable
copy of itself.
The study of the properties of the complete set of such operations forms
one application of group theory. Though this is the aspect of most interest to
the physical scientist, group theory itself is a much larger subject and of great
importance in its own right. Consequently we leave until the next chapter any
direct applications of group theoretical results and concentrate on building up
the general mathematical properties of groups.
28.1 Groups
As an example of symmetry properties, let us consider the sets of operations,
such as rotations, reflections, and inversions, that transform physical objects, for
example molecules, into physically indistinguishable copies of themselves, so that
only the labelling of identical components of the system (the atoms) changes in
the process. For differently shaped molecules there are different sets of operations,
but in each case it is a well-defined set, and with a little practice all members of
each set can be identified.
As simple examples, consider (a) the hydrogen molecule, and (b) the ammonia
molecule illustrated in figure 28.1. The hydrogen molecule consists of two atoms
H of hydrogen and is carried into itself by any of the following operations:
(i) any rotation about its long axis;
(ii) rotation through π about an axis perpendicular to the long axis and
passing through the point M that lies midway between the atoms;
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GROUP THEORY
N
H
M
H
H
H
(a)
(b)
H
Figure 28.1 (a) The hydrogen molecule, and (b) the ammonia molecule.
(iii) inversion through the point M;
(iv) reflection in the plane that passes through M and has its normal parallel
to the long axis.
These operations collectively form the set of symmetry operations for the hydrogen molecule.
The somewhat more complex ammonia molecule consists of a tetrahedron with
an equilateral triangular base at the three corners of which lie hydrogen atoms
H, whilst a nitrogen atom N is sited at the fourth vertex of the tetrahedron. The
set of symmetry operations on this molecule is limited to rotations of π/3 and
2π/3 about the axis joining the centroid of the equilateral triangle to the nitrogen
atom, and reflections in the three planes containing that axis and each of the
hydrogen atoms in turn. However, if the nitrogen atom could be replaced by a
fourth hydrogen atom, and all interatomic distances equalised in the process, the
number of symmetry operations would be greatly increased.
Once all the possible operations in any particular set have been identified, it
must follow that the result of applying two such operations in succession will be
identical to that obtained by the sole application of some third (usually different)
operation in the set – for if it were not, a new member of the set would have
been found, contradicting the assumption that all members have been identified.
Such observations introduce two of the main considerations relevant to deciding whether a set of objects, here the rotation, reflection and inversion operations,
qualifies as a group in the mathematically tightly defined sense. These two considerations are (i) whether there is some law for combining two members of the set,
and (ii) whether the result of the combination is also a member of the set. The
obvious rule of combination has to be that the second operation is carried out
on the system that results from application of the first operation, and we have
already seen that the second requirement is satisfied by the inclusion of all such
operations in the set. However, for a set to qualify as a group, more than these
two conditions have to be satisfied, as will now be made clear.
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28.1 GROUPS
28.1.1 Definition of a group
A group G is a set of elements {X, Y , . . . }, together with a rule for combining
them that associates with each ordered pair X, Y a ‘product’ or combination law
X • Y for which the following conditions must be satisfied.
(i) For every pair of elements X, Y that belongs to G, the product X • Y also
belongs to G. (This is known as the closure property of the group.)
(ii) For all triples X, Y , Z the associative law holds; in symbols,
X • (Y • Z) = (X • Y ) • Z.
(28.1)
(iii) There exists a unique element I, belonging to G, with the property that
I •X =X =X•I
(28.2)
for all X belonging to G. This element I is known as the identity element
of the group.
(iv) For every element X of G, there exists an element X −1 , also belonging to
G, such that
X −1 • X = I = X • X −1 .
(28.3)
X −1 is called the inverse of X.
An alternative notation in common use is to write the elements of a group G
as the set {G1 , G2 , . . . } or, more briefly, as {Gi }, a typical element being denoted
by Gi .
It should be noticed that, as given, the nature of the operation • is not stated. It
should also be noticed that the more general term element, rather than operation,
has been used in this definition. We will see that the general definition of a
group allows as elements not only sets of operations on an object but also sets of
numbers, of functions and of other objects, provided that the interpretation of •
is appropriately defined.
In one of the simplest examples of a group, namely the group of all integers
under addition, the operation • is taken to be ordinary addition. In this group the
role of the identity I is played by the integer 0, and the inverse of an integer X is
−X. That requirements (i) and (ii) are satisfied by the integers under addition is
trivially obvious. A second simple group, under ordinary multiplication, is formed
by the two numbers 1 and −1; in this group, closure is obvious, 1 is the identity
element, and each element is its own inverse.
It will be apparent from these two examples that the number of elements in a
group can be either finite or infinite. In the former case the group is called a finite
group and the number of elements it contains is called the order of the group,
which we will denote by g; an alternative notation is |G| but has obvious dangers
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GROUP THEORY
if matrices are involved. In the notation in which G = {G1 , G2 , . . . , Gn } the order
of the group is clearly n.
As we have noted, for the integers under addition zero is the identity. For
the group of rotations and reflections, the operation of doing nothing, i.e. the
null operation, plays this role. This latter identification may seem artificial, but
it is an operation, albeit trivial, which does leave the system in a physically
indistinguishable state, and needs to be included. One might add that without it
the set of operations would not form a group and none of the powerful results
we will derive later in this and the next chapter could be justifiably applied to
give deductions of physical significance.
In the examples of rotations and reflections mentioned earlier, • has been taken
to mean that the left-hand operation is carried out on the system that results
from application of the right-hand operation. Thus
Z =X•Y
(28.4)
means that the effect on the system of carrying out Z is the same as would
be obtained by first carrying out Y and then carrying out X. The order of the
operations should be noted; it is arbitrary in the first instance but, once chosen,
must be adhered to. The choice we have made is dictated by the fact that most
of our applications involve the effect of rotations and reflections on functions of
space coordinates, and it is usual, and our practice in the rest of this book, to
write operators acting on functions to the left of the functions.
It will be apparent that for the above-mentioned group, integers under ordinary
addition, it is true that
Y •X =X•Y
(28.5)
for all pairs of integers X, Y . If any two particular elements of a group satisfy
(28.5), they are said to commute under the operation •; if all pairs of elements in
a group satisfy (28.5), then the group is said to be Abelian. The set of all integers
forms an infinite Abelian group under (ordinary) addition.
As we show below, requirements (iii) and (iv) of the definition of a group
are over-demanding (but self-consistent), since in each of equations (28.2) and
(28.3) the second equality can be deduced from the first by using the associativity
required by (28.1). The mathematical steps in the following arguments are all
very simple, but care has to be taken to make sure that nothing that has not
yet been proved is used to justify a step. For this reason, and to act as a model
in logical deduction, a reference in Roman numerals to the previous result,
or to the group definition used, is given over each equality sign. Such explicit
detailed referencing soon becomes tiresome, but it should always be available if
needed.
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28.1 GROUPS
Using only the first equalities in (28.2) and (28.3), deduce the second ones.
Consider the expression X −1 • (X • X −1 );
(ii)
(iv)
X −1 • (X • X −1 ) = (X −1 • X) • X −1 = I • X −1
(iii) −1
= X .
But X
−1
(28.6)
belongs to G, and so from (iv) there is an element U in G such that
U • X −1 = I.
(v)
Form the product of U with the first and last expressions in (28.6) to give
(v)
U • (X −1 • (X • X −1 )) = U • X −1 = I.
(28.7)
Transforming the left-hand side of this equation gives
(ii)
U • (X −1 • (X • X −1 )) = (U • X −1 ) • (X • X −1 )
(v)
= I • (X • X −1 )
(iii)
= X • X −1 .
(28.8)
Comparing (28.7), (28.8) shows that
X • X −1 = I,
(iv)
i.e. the second equality in group definition (iv). Similarly
(iv)
(ii)
X • I = X • (X −1 • X) = (X • X −1 ) • X
(iv)
= I •X
(iii)
= X.
(iii )
i.e. the second equality in group definition (iii). The uniqueness of the identity element I can also be demonstrated rather than
assumed. Suppose that I , belonging to G, also has the property
I • X = X = X • I
for all X belonging to G.
Take X as I, then
I • I = I.
(28.9)
Further, from (iii ),
X =X•I
for all X belonging to G,
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GROUP THEORY
and setting X = I gives
I = I • I.
(28.10)
It then follows from (28.9), (28.10) that I = I , showing that in any particular
group the identity element is unique.
In a similar way it can be shown that the inverse of any particular element
is unique. If U and V are two postulated inverses of an element X of G, by
considering the product
U • (X • V ) = (U • X) • V ,
it can be shown that U = V . The proof is left to the reader.
Given the uniqueness of the inverse of any particular group element, it follows
that
(U • V • · · · • Y • Z) • (Z −1 • Y −1 • · · · • V −1 • U −1 )
= (U • V • · · · • Y ) • (Z • Z −1 ) • (Y −1 • · · · • V −1 • U −1 )
= (U • V • · · · • Y ) • (Y −1 • · · · • V −1 • U −1 )
..
.
= I,
where use has been made of the associativity and of the two equations Z • Z −1 = I
and I • X = X. Thus the inverse of a product is the product of the inverses in
reverse order, i.e.
(U • V • · · · • Y • Z)−1 = (Z −1 • Y −1 • · · · • V −1 • U −1 ).
(28.11)
Further elementary results that can be obtained by arguments similar to those
above are as follows.
(i) Given any pair of elements X, Y belonging to G, there exist unique
elements U, V , also belonging to G, such that
X•U =Y
V •X =Y.
and
−1
−1
Clearly U = X • Y , and V = Y • X , and they can be shown to be
unique. This result is sometimes called the division axiom.
(ii) The cancellation law can be stated as follows. If
X•Y =X•Z
for some X belonging to G, then Y = Z. Similarly,
Y •X =Z •X
implies the same conclusion.
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28.1 GROUPS
M
L
K
Figure 28.2 Reflections in the three perpendicular bisectors of the sides of
an equilateral triangle take the triangle into itself.
(iii) Forming the product of each element of G with a fixed element X of G
simply permutes the elements of G; this is often written symbolically as
G • X = G. If this were not so, and X • Y and X • Z were not different
even though Y and Z were, application of the cancellation law would lead
to a contradiction. This result is called the permutation law.
In any finite group of order g, any element X when combined with itself to
form successively X 2 = X • X, X 3 = X • X 2 , . . . will, after at most g − 1 such
combinations, produce the group identity I. Of course X 2 , X 3 , . . . are some of
the original elements of the group, and not new ones. If the actual number of
combinations needed is m − 1, i.e. X m = I, then m is called the order of the element
X in G. The order of the identity of a group is always 1, and that of any other
element of a group that is its own inverse is always 2.
Determine the order of the group of (two-dimensional) rotations and reflections that take
a plane equilateral triangle into itself and the order of each of the elements. The group is
usually known as 3m (to physicists and crystallographers) or C3v (to chemists).
There are two (clockwise) rotations, by 2π/3 and 4π/3, about an axis perpendicular to
the plane of the triangle. In addition, reflections in the perpendicular bisectors of the three
sides (see figure 28.2) have the defining property. To these must be added the identity
operation. 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 original
situation. Thus each rotation element of the group has order 3, and each reflection element
has order 2. A so-called cyclic group is one for which all members of the group can be
generated from just one element X (say). Thus a cyclic group of order g can be
written as
2
1
G = I, X, X 2 , X 3 , . . . , X g−1 .
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GROUP THEORY
It is clear that cyclic groups are always Abelian and that each element, apart
from the identity, has order g, the order of the group itself.
28.1.2 Further examples of groups
In this section we consider some sets of objects, each set together with a law of
combination, and investigate whether they qualify as groups and, if not, why not.
We have already seen that the integers form a group under ordinary addition,
but it is immediately apparent that (even if zero is excluded) they do not do
so under ordinary multiplication. Unity must be the identity of the set, but the
requisite inverse of any integer n, namely 1/n, does not belong to the set of
integers for any n other than unity.
Other infinite sets of quantities that do form groups are the sets of all real
numbers, or of all complex numbers, under addition, and of the same two sets
excluding 0 under multiplication. All these groups are Abelian.
Although subtraction and division are normally considered the obvious counterparts of the operations of (ordinary) addition and multiplication, they are not
acceptable operations for use within groups since the associative law, (28.1), does
not hold. Explicitly,
X − (Y − Z) = (X − Y ) − Z,
X ÷ (Y ÷ Z) = (X ÷ Y ) ÷ Z.
From within the field of all non-zero complex numbers we can select just those
that have unit modulus, i.e. are of the form eiθ where 0 ≤ θ < 2π, to form a
group under multiplication, as can easily be verified:
= ei(θ1 +θ2 )
eiθ1 × eiθ2
=1
ei0
ei(2π−θ) × eiθ = ei2π ≡ ei0 = 1
(closure),
(identity),
(inverse).
Closely related to the above group is the set of 2 × 2 rotation matrices that take
the form
cos θ − sin θ
M(θ) =
sin θ cos θ
where, as before, 0 ≤ θ < 2π. These form a group when the law of combination
is that of matrix multiplication. The reader can easily verify that
M(θ)M(φ) = M(θ + φ) (closure),
(identity),
M(0) = I2
M(2π − θ) = M−1 (θ)
(inverse).
Here I2 is the unit 2 × 2 matrix.
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