2
DEFINITIONS AND THEOREMS
GROUP THEORY
2.1 THE DEFINING PROPERTIES OF A GROUP
Agroup is a collection of elements
rules. VWe need not specify what thethat are interrelated according to certain
significance to them in order to iscusselements
the
are or attribute any physicai
group which they constitute. In
this book, of course, we shall be
sets of symmetry operations that may concerned with the groups formed by the
be
but the basic definitions and theorems ofcarried out on molecules or crystals.
group theory are far more general.
In order for any set of elements to form a mathematical group, the following
conditions or rules must be satisfied.
1. The product of any two elements in the group and the square of each
element must be an element in the group. In order for this condition to have
meaning, we must, of course, have agreed on what we mean by the terms
"multiply" and "product." They need not mean what they do in ordinary
algebra and arithmetic. Perhaps we might say "combine" instead of multi
ply" and "combination" instead of "product" in order to avoid unnecessary
and perhaps inc¡rrect connotations. Let us not yet commit ourselves to any
particular law of combination but merely say that, if A and B are two elements
of a group, we indicate that we are combining them by simply writing AB or
BA. Now immediately the question arises if it makes any difference whether
we write AB or BA. In ordinary algebra it does not, and we say that multr
the
plhcation is commutative, that is xy = yx, or 3 X 6 = 6 X 3. In group
AB may give
ory, the commnutative law does not in general hold. Thus
group.
while BA may give D, where Cand D are two more elements in the
DEFINITIONS AND THEOREMS OF GROUP THEORY
Lnere are some groups, however, in which combination is commutative, and
Sucn groups are called Abelian groups. Because of the fact that muiupheao
S not in general commutative. it is sometimes convenient to have a means
of stating whether an element Bis to be multiplied by Ain the sense Ab Or
BA. In the first case we can say that Bis let-multiplied by A, and in tne
second case that Bis right-multiplied by A.
2. One element in the groun mut commute with all others and leave them
unchanged. It is customary to designate this element with the leter E, and
writing
it is usually called the identity element. Svmbolically we define it by
EX = XE = X.
asSOciative law of multiplicationmust hold. This is expressed in the
3. The
following equality:
A(BC) = (AB)C
order BC and then combine
B with Cin the
In plain words, we may combine CombineA with Bin the
or we may
this product, S, with Ain the order AS,which we then combine with Cinthe
order AB, obtaining a product, say R, either way. In general, of course,
order RC and get the same final product continued product of any number
for the
the associative property must hold
of elements, namely, (AB)C(DE)\(FG)H *"
A(BCEDE)XFG)H =
(AB)(CD)(EF)\GH) = element of the
which is also an
Every element must have a reciprocal,the element S if RS = SR = E,
4.
element R is the reciprocal of reciprocal of S, then S is the
group. The
identity. Obviously, if R is the
where E is the reciprocal.
of R. Also, E is its own
reciprocal which
theorem concerningreciprocals
we shall prove asmall
Atthispoint
later. The rule is
will be of use equalto theproduct
elementsis
productoftwoormore
ofa
Thereciprocal in reverseorder.
reciprocals,
ofthe
This means that C-'B-'A-!
Y-'X- ..
XY)-! = be
(ABC"*. product, butitwill
their
prove thisfora ternarygroupelements,
simplicitywe shall B, andCare
A,
PROOF. For true
generally. If element,namely,
is group
obvious that it mustalsobea
product,sayD, ABC = D
8 PRINCIPLES
If now we right-multiply each side of this equation by C'B-'A-! we obtain
ABCC-'B-'4-! = DC-'B-A-!
DC-B-4-1
ABEB-'A -I =
E = DC-B-'A-1
reciprocal ofD, and
C-'B-'4- isthe
since
Since Dtimes C-'B-'A- = E,
D= ABC, we have
C-'B-'4-1
=
Dl= (ABC)-!
which proves the above rule.
OF GROUPS
2.2 SOME EXAMPLES will be Con-
until we reach Chapter 11, significance
book, The
Most of our attention in this groupcalled apoint group. not detain us
need
centrated on a type ofsymmetry group"
group."
and "point number of elements, but
of these terms, "symmetry a finite
Most of them containinfinite. The number of elements
(see Chapter 3).
here molecules belong) areconventional symbol for the order
two (to which linearcalled its order, and the an infinite group
in afinite group isthe above defining rules, we may consider
is h. To illustrate positive and
and then some finite groups.may take allof the integers, both
we algebraic
As an infinite group take as our law of combination the ordinary
negative, and zero. If we may be
addition, then rule 1is satisfied. Clearly, any integer
group since
process of others. Note that we have an Abelian
obtained by adding two since 0 +
The identity of the group is 0, since, for
the order of addition is immaterial.
associative law of combination holds,
n=n + 0= n. Also, the
example, ((+3) + (-7)) + (+1043) = (+3) + [(-7) + (+1043). The
(-n), since (+n) + (-n) = 0.
reciprocal of any element, n, is
Group Multiplication Tables
If we have a complete and nonredundant list of the h elements of afinite
group and we know what all of the possible products (there are h) are, then
the group is completely and uniquely defined-at least in an abstract sense.
The foregoing information can be presented most conveniently in the form
of the group multiplication table. This consists of h rows and h columns. Each
column is labeled with a group element, and so is each row. The entry in the
table under a given column and along a given row is the product of the
elements which head that column and that row. Because multiplication isin
9
DEFINITIONS AND THEOREMS OF GROUP THEORY
general not commutative, we must have an agreed upon and consistent rule
for the order of we shall take the factors
in the
order: multiplication.
(oolumn element) x (rowArbitrarily.
element Thus at the intersection of the
column labeled by which
Xand the row labeled by Y we find the element,
is the product XY.
,oW prove an important theorem about group multiplication tables,
called the rearrangement theorem.
Each row and each column in the group multiplication table lists
each of
that no lwo ro
ne group elements once and only once. From this. it follows
identical. Thus each roW ana
may be ldentical nor may anytwo columns be
each column is a rearranged list of the group elements.
A,. The
PROOF. Let the group consist of the h elements E, Az, A3,
elements in a given row, say the nth row, are
EA, A,An, ,A,An . ..,A,A,
elements, A, and A. for instance, are the same, no tWoare
Since n0 two group The h entries in the nth row
products, A,A, and A,A,, can be the same. elements, each of them must be
h group
all diferent. Since there are onlyargument can obviously be adaptedtothÃ
The
present once and only once.
columns.
3
Groups of Orders 1, 2, and abstract groups of low order,
possible formally
systematically examine the is,of course,
Let us nowmultiplication tables todefine them. There alone. Thereis only
using their identity element
order 1, which consists of the
following multiplication table and
agroup of has the
possible group of order 2. It
one
willbe designated G
GaE A
E E A
A|A E
have to be, in part,
multiplication table will
order 3,the
For agroup of
asfollows: EA B
B
E A
AA
BIB
10 PRINCIPLES
There is then only one way to complete the table. Either AA = Bor AA
E. IfAA = E,then BB = E and we would augment the table to give
E A B
E A B
AA E
B B E
But then we can get no further, since we would have to accept BA = Aand
AB = Ain order to complete the last column and the last row,
thus repeating Ain both the second column and the second row.respectively.
The alter.
native, AA =B, leads unambiguously to the following table:
G, EA B
E E A B
A A B E
BIB E A
CyclicGroups
G is the simplest, nontrivial member of an important set of groups, the cyclic
consider
groups. We note that AA = B, while AB(=AAA) = E. Thus we can
the entire group to be generated by taking the element A and its powers,
A{= B) and A=E). In general, the cyclic group ofshallorder h is de fined as
presently examine
an element Xand all of its h powers up to X= E. We
several other cyclic groups. An important property of cyclic groups is that
commutative. This must be
they are Abelian, that is, all multiplications arethe form X, X", and so on,
are all of
So, since the various group elements
and n.
and, clearly, X"X" = X"X" for all m
Groupsof Order 4
many groups of order 4 there are and what their
To continue, we ask how will be acyclic group of order
multiplication table(s) willbe. Obviously, there
relations.
4. Let us employ the
X =A X = C
X=B X = E
as
multiplication table, in the usual format, is
From this we find that the
follows:
DEFINITIONS AND THEOREMS OF GROUP THEOKT 11
G) E A BC
E E A B C
A A B CE
B B C A
C E A B
That there is a second type of G, group, G), is fairly obvious. We note
that for G onlyone element, namely B, is its own inverse. Supp0Se, instead,
we assume that each of two elements. A and B. is its own inverse. Ne shall
then have no choice but to also make Cits own inverse, since each of the
four E's in the table must lie in a diferent row and column, Thus, we woula
obtain
E A B C
E A B C
A A E
B B E
CIC E
Amoment's consideration will show that there is only one way to complete
this table:
G E A BC
E E A B
A A E C B
B C E A
C BA E
two
It is also clear that there are no other possibilities.* Thus, there are be
considered to
groups of order 4, namely G and GP, which may be
defined bytheir multiplication tables.
Groups of Orders 5 and 6
group of
It is left as an exercise (Exercise 2.2) to show that there is only one of
order 5. Similarly, a systematic examination of the possibilities for groups
than E) is its own inverse and let that
If we make up a table in which only one element (other not inventing a diferent G..
element be A or C instead of B as in the G table given, we are
the group elements.
We are only permuting the arbitrary symbols for
12 PRINCIPLES
order 6is also left as an exercise(Exercise 2.9). To provide
for several topics that we shall take up next, the
of the groups of order 6 is given. multipliciatl uiostnrattableive mafor terioneal
G E A B C D
E E A B C D
A E D F B C
B B F E D A
D F E A B
D D CA B E
F F B C A E D
2.3 SUBGROUPS
Inspection of the multiplication table for the group G will show that within
this gToup of order6 there are
smaller
group of order 1. This will, of course, be groups. The identity E in itself is a
anontrivial nature are the true in any group and is
groups of order 2, namely, E, A; E, B; trivial. Of
the group of order 3, namely, E, D, F. E, C; and
The last should be
F, D' = DF = FD = E.recognized
as the cyclic group G3, since D' = also
to the main point, smaller But to return
groups
are called subgroups. There are, of that may be found within a larger group
other than the trivial one of E itself. course, groups that have no subgroups
Let us now consider whether there are
any restrictions on the nature of
subgroups, restrictions that are logical consequences of the general definition
of a group and not of any additional or special
group. We may note that the orders of the group characteristics of a particular
Cy
6and 1,2, 3; in short, the orders of the subgroups are alland its subgroups are
of the main group. We shall now prove the following theorem:
factors of the order
The order of any subgroup g of agroup of order h mustbe a divisor of h.
In other words, hlg = k where k is an integer.
PROOF. Suppose that the set of gelements, A,, Az, As, ..., A,, forms a
subgroup. Now let us take another element B in the group which is not a
member of this subgroup and form all of the g products: BA,, BA,, .
DA,. No on of these products can be in the subgroup. If, for example,
BA, = A,
DEFINITIONS AND THEOREMS OF GROUP THEORY 13
then, if we take the
equality, we obtain reciprocal A,. perhaps A,and right-mutiply the abov
of
BA,A, = A,As
BE = A,As
B = AA,
But this contradicts Our assumption that B is not a member of the subgroup
A, Az,....,A, since AAs can only be one of the A,. Hence, if all the
oroducts BA, are in the large group in addition to the A. themselves, there
are at least 2g members of the group. If h> 2g, we can choose still another
element of the group, namely C, which is neither one of the A, nor one or
the BAj, and on multiplying the A,by Cwe willobtain g more elements, all
members of the main group, but none members of the A,or of the BA,sets.
Thus we now know that hmust be at least equal to 3g. Eventually, however,
we must reach the point where there are no more elements by which we can
multiply the A, that are not among the sets A,, BA,, CA,, and so forth, already
obtained. Suppose after having found k such elements, we reach the point
where there are no more. Then h = kg, where k is an integer, and hlg =
L which is what we set out to prove.
Although we have shown that the order of any subgroup, 8, must be a
divisor of h, we have not proved the converse, namely, that there are subgrOups
of all orders that are divisors of h, and, indeed, this is not in general true.
than one
Moreover, as our illustrative group proves, there can be more
subgroup of a given order.
2.4 CLASSES
various smaller
We have seen that in agiven group it may be possible to select
in themselves
sets of elements, each such set including E, however, which are
may be
groups. There is another way in which the elements of a group Before defining
Separated into smaller sets, and such sets are called classes.
a class we must consider an operation known as similarity
transformation.
equal to
IiAand Xare twWo elements of a group, thenX"AX will be
SOme element of the group, say B. We have
B = X-AX
We express this relation in words by saying that Bis the similarity transform
otA by X. We also say that A and Bare conjugate. The following properties
of conjugate elements are important.
PRINCIPLES
14
(i) Every element is conjugate with itself. This means that if
any particular element Ait must be possible to find at
least one choo*
we
such that
If we left-multiply by A we obtain
A = X-'AX element x
A-'A = E = A-X-'AX =
which can hold only if Aand X commute. Thus the (XA)(AX)
be E, andit may be any other element that
A.
element
commutes with X may
the chosen always
if
(i) If Ais conjugate with B, then Bis
conjugate with A. This
element,
means that
A = X-'BX
then there must be some
element Y in the group such that
B= Y-AY
That this must be so is easily
cations, namely, proved by carrying out appropriate multipli
XAX-1 =
XX-1 BXX-1 = B
Thus, ifY =X- (and thus also Y-l = X), we
have
B = Y-'AY
and this must be possible, since any
say Y.
element, say X, must have an inverse,
(ii) If A is conjugate with B and C, then B and C are conjugate with each
other. The proof of this should be easy to work out from the foregoing
discussion and is left as an exercise.
We may now define a class of elenments.
A complete set of elements that are conjugate to one another is called a class
of the group.
In order to determine the classes within any particular group we can begin
the elements
with one element and work out all of its transforms, using all is not one
second element, which
in the group, including itself, then take a and determine allits transforms,
first,
of those found to be conjugate to thegroup have been placed in one class or
and so on until all elements in the
another. the group G. All of the results given
Let us illustrate this procedure with with E.
themultiplication table. Let us start
Delow may be verified by using
A
15
UEHINITIONS AND THEOREMSs OF GROUP THEORY
E-EE EEE = L
A-'EA = A'AE = E
B-'EB = B-BE = E
Thus E must constitute by itself a class. of order 1. since it is not
with any other element. This will, of course, be true in any conjugate
group. To conunu
E-AE = A
A-'AA = A
B-'AB = C
C-'AC = B
D-'AD = B
F-'AF = C
Thus the elements A, B, and C are all conjugate and are therefore
members
of the same class. It is left for the reader to show that all of the transforms
of B and Care either A, B, or C. Thus A. B. and C are in fact the only
members of the class.
Continuing we have
E-'DE = D
A -'DA = F
B-DB = F
C-DC = F
D-'DD = D
F-DF D
It will also be found that every transform of Fis either Dor F. Hence, D
and F constitute a class of order 2.
It will be noted that the classes have orders 1, 2, and 3, which are all
factors of the group order, 6. It can be proved, by a method similaf to that
used in connection with the orders of subgroups, that the following theorem
is true:
The orders of all classes must be integral factors of the order of the group.
16 PRINCIPLES
groupthe classes have
We shall see later (Section 3.13) that in asymmetry
useful geometrical significance.
EXERCISES itself.
class by
element isin a
each is a prime
2.1. Prove that in any Abelian group, order h, when h
group of
can be only one
2.2. Show that there of order 5.
number. cyclic group
table for the possible.
multiplication
2.3. Write down the error that no other one is E?
Show by trial and A' = B²
group in which and add another
not have a group G:
2.4. Why can we tablefor what multiplication
multiplication and B,
2.5. If westart with the commutes with both A
element., C, which there must
table do we end
up with? ,X(=E), orderh.
cyclicgroup, X, X", X°,
integraldivisorofthe
for anycorresponding to each
2.6. Show that
onesubgroup can and
be
Give an example. order 8 as you
groups of
noncyclic
Invent many diferent
asmultiplication table for each. down into
2.7. how it breaks
give the of order 8, show
groups
each of the
For
2.8. subgroupsand classes. groups of order 6 besides
all other group
multiplication table for require you to show that a
the impossible.
2,9, Derive shown in the text. This will its own inverse is
the one every element is what
of order 6in
which
group of order 6, show
G", G and the cyclic
groups
2.10. For the subgroups each one has.
classes and
