II. The Deﬁnition of a Group.

Chapter II.
The Deﬁnition of a Group.

In the present chapter we shall enter on our main subject and we shall begin with deﬁnitions, explanations and examples of what is meant by a group.

Deﬁnition. Let

A, B, C, \dots

represent a set of operations, which can be performed on the same object or set of objects. Suppose this set of operations has the following characteristics.

( $α$ ) The operations of the set are all distinct, so that no two of them produce the same change in every possible application.

( $β$ ) The result of performing successively any number of operations of the set, say $A$ , $B$ , …, $K$ , is another deﬁnite operation of the set, which depends only on the component operations and the sequence in which they are carried out, and not on the way in which they may be regarded as associated. Thus $A$ followed by $B$ and $B$ followed by $C$ are operations of the set, say $D$ and $E$ ; and $D$ followed by $C$ is the same operation as $A$ followed by $E$ .

( $γ$ ) $A$ being any operation of the set, there is always another operation $A_{- 1}$ belonging to the set, such that $A$ followed by $A_{- 1}$ produces no change in any object.

The operation $A_{- 1}$ is called the inverse of $A$ .

The set of operations is then said to form a Group.

From the deﬁnition of the inverse of $A$ given in ( $γ$ ), it follows directly that $A$ is the inverse of $A_{- 1}$ . For if $A$ changes any object $Ω$ into $Ω'$ , $A_{- 1}$ must change $Ω'$ into $Ω$ . Hence $A_{- 1}$ followed by $A$ leaves $Ω'$ , and therefore every object, unchanged.

The operation resulting from the successive performance of the operations $A$ , $B$ , …, $K$ in the sequence given is denoted by the symbol $A B \dots K$ ; and if $Ω$ is any object on which the operations may be performed, the result of carrying out this compound operation on $Ω$ is denoted by $Ω \cdot A B \dots K$ .

If the component operations are all the same, say $A$ , and $r$ in number, the abbreviation $A^{r}$ will be used for the resultant operation, and it will be called the $r$ th power of $A$ .

Deﬁnition. Two operations, $A$ and $B$ , are said to be permutable when $A B$ and $B A$ are the same operation.

13. If $A B$ and $A C$ are the same operation, so also are $A_{- 1} A B$ and $A_{- 1} A C$ . But the operation $A_{- 1} A$ produces no change in any object and therefore $A_{- 1} A B$ and $B$ , producing the same change in every object, are the same operation. Hence $B$ and $C$ are the same operation.

This is expressed symbolically by saying that, if

A B = A C,

then

B = C;

the sign of equality being used to imply that the symbols represent the same operation.

In a similar way, if

B A = C A,

it follows that

B = C .

From conditions ( $β$ ) and ( $γ$ ), $A A_{- 1}$ must be a deﬁnite operation of the group. This operation, by deﬁnition, produces no change in any possible object, and it must, by condition ( $α$ ), be unique. It is called the identical operation. If it is represented by $A_{0}$ and if $A$ be any other operation, then

A_{0} A = A = A A_{0},

and for every integer

r

A_{0}^{r} = A_{0} .

Hence $A_{0}$ may, without ambiguity, be replaced by $1$ , wherever it occurs.

14. The number of distinct operations contained in a group may be either ﬁnite or inﬁnite. When the number is inﬁnite, the group may contain operations which produce an inﬁnitesimal change in every possible object or operand.

Thus the totality of distinct displacements of a rigid body evidently forms a group, for they satisfy conditions ( $α$ ), ( $β$ ) and ( $γ$ ) of the deﬁnition. Moreover this group contains operations of the kind in question, namely inﬁnitesimal twists; and each operation of the group can be constructed by the continual repetition of a suitably chosen inﬁnitesimal twist.

Next, the set of translations, that arise by shifting a cube parallel to its edges through distances which are any multiples of an edge, forms a group containing an inﬁnite number of operations; but this group contains no operation which eﬀects an inﬁnitesimal change in the position of the cube.

As a third example, consider the set of displacements by which a complete right circular cone is brought to coincidence with itself. It consists of rotations through any angle about the axis of the cone, and rotations through two right angles about any line through the vertex at right angles to the axis. Once again this set of displacements satisﬁes the conditions ( $α$ ), ( $β$ ) and ( $γ$ ) of the deﬁnition and forms a group.

This last group contains inﬁnitesimal operations, namely rotations round the axis through an inﬁnitesimal angle; and every ﬁnite rotation round the axis can be formed by the continued repetition of an inﬁnitesimal rotation. There is however in this case no inﬁnitesimal displacement of the group by whose continued repetition a rotation through two right angles about a line through the vertex at right angles to the axis can be constructed. Of these three groups with an inﬁnite number of operations, the ﬁrst is said to be a continuous group, the second a discontinuous group, and the third a mixed group.

Continuous groups and mixed groups lie entirely outside the plan of the present treatise; and though, later on, some of the properties of discontinuous groups with an inﬁnite number of operations will be considered, such groups will be approached from a point of view suggested by the treatment of groups containing a ﬁnite number of operations. It is not therefore necessary here to deal in detail with the classiﬁcation of inﬁnite groups which is indicated by the three examples given above; and we pass on at once to the case of groups which contain a ﬁnite number only of distinct operations.

15. Deﬁnition. If the number of distinct operations contained in a group be ﬁnite, the number is called the order of the group.

Let $S$ be an operation of a group of ﬁnite order $N$ . Then the inﬁnite series of operations

S, S^{2}, S^{3}, \dots

must all be contained in the group, and therefore a ﬁnite number of them only can be distinct. If

S^{m + 1}

is the ﬁrst of the series which is the same as

S

, and if

S_{- 1}

is the operation inverse to

S

, then

S^{m + 1} S_{- 1} = S S_{- 1} = 1,

S^{m} = 1 .

Exactly as in § 8, it may be shewn that, if

S^{μ} = 1,

μ

must be a multiple of

m

, and that the operations

S

S^{2}

, …,

S^{m - 1}

are all distinct.

Since the group contains only $N$ distinct operations, $m$ must be equal to or less than $N$ . It will be seen later that, if $m$ is less than $N$ , it must be a factor of $N$ .

The integer $m$ is called the order of the operation $S$ . The order $m'$ of the operation $S^{x}$ is the least integer for which

S^{x m'} = 1,

that is, for which

x m' \equiv 0 (mod m) .

Hence, if $g$ is the greatest common factor of $x$ and $m$ ,

m' = \frac{m}{g};

and, if

m

is prime, all the powers of

S

, whose indices are less than

m

, are of order

m

Since

S^{x} S^{m - x} = S^{m} = 1, (x < m),

and

S^{x} {(S^{x})}_{- 1} = 1,

it follows that

{(S^{x})}_{- 1} = S^{m - x} .

If now a meaning be attached to $S^{- x}$ , by assuming that the equation

S^{x + y} = S^{x} S^{y}

holds when either

x

y

is a negative integer, then

S^{m - x} = S^{m} S^{- x} = S^{- x},

and

{(S^{x})}_{- 1} = S^{- x},

so that

S^{- x}

denotes the inverse of the operation

S^{x}

Ex. If $S_{a}$ , $S_{b}$ , …, $S_{c}$ , $S_{d}$ are operations of a group, shew that the operation inverse to $S_{a}^{α} S_{b}^{β} \dots S_{c}^{γ} S_{d}^{δ}$ is $S_{d}^{- δ} S_{c}^{- γ} \dots S_{b}^{- β} S_{a}^{- α}$ .

16. If

1, S_{1}, S_{2}, \dots, S_{N - 1}

are the

N

operations of a group of order

N

, the set of

N

operations

S_{r}, S_{r} S_{1}, S_{r} S_{2}, \dots, S_{r} S_{N - 1}

are (§ 13) all distinct; and their number is equal to the order of the group. Hence every operation of the group occurs once and only once in this set.

Similarly every operation of the group occurs once and only once in the set

S_{r}, S_{1} S_{r}, S_{2} S_{r}, \dots, S_{N - 1} S_{r} .

Every operation of the group can therefore be represented as the product of two operations of the group, and either the ﬁrst factor or the second factor can be chosen at will.

A relation of the form

S_{p} = S_{q} S_{r}

between three operations of the group will not in general involve any necessary relation between the order of

S_{p}

and the orders of

S_{q}

and

S_{r}

. If however the two latter are permutable, the relation requires that, for all values of

x

S_{p}^{x} = S_{q}^{x} S_{r}^{x};

and in that case the order of

S_{p}

is the least common multiple of the orders of

S_{q}

and

S_{r}

Suppose now that $S$ , an operation of the group, is of order $m n$ , where $m$ and $n$ are relatively prime. Then we may shew that, of the various ways in which $S$ may be represented as the product of two operations of the group, there is just one in which the operations are permutable and of orders $m$ and $n$ respectively.

Thus let

S^{n} = M,

and

S^{m} = N,

so that

M

N

are operations of orders

m

and

n

. Since

S^{m}

and

S^{n}

are permutable, so also are

M

and

N

, and powers of

M

and

N

If $x_{0}$ , $y_{0}$ are integers satisfying the equation

x n + y m = 1,

every other integral solution is given by

x = x_{0} + t m, y = y_{0} - t n,

where

t

is an integer.

Now

M^{x} N^{y} = S^{x n + y m} = S;

and since

x

and

m

are relatively prime, as also are

y

and

n

M^{x}

and

N^{y}

are permutable operations of orders

m

and

n

, so that

S

is expressed in the desired form.

Moreover, it is the only expression of this form; for let

S = M_{1} N_{1},

where

M_{1}

and

N_{1}

are permutable and of orders

m

and

n

Then $S^{n} = M_{1}^{n}$ , since $N_{1}^{n} = 1$ .

Hence

M_{1}^{n} = M,

M_{1}^{x n} = M^{x},

M_{1}^{1 - y m} = M^{x} .

But

M_{1}^{m} = 1

, and therefore

M_{1}^{- y m} = 1

; hence

M_{1} = M^{x} .

In the same way it is shewn that

N_{1}

is the same as

N^{y}

. The representation of

S

in the desired form is therefore unique.

17. Two given operations of a group successively performed give rise to a third operation of the group which, when the operations are of known concrete form, may be determined by actually carrying out the two given operations. Thus the set of ﬁnite rotations, which bring a regular solid to coincidence with itself, evidently form a group; and it is a purely geometrical problem to determine that particular rotation of the group which arises from the successive performance of two given rotations of the group.

When the operations are represented by symbols, the relation in question is represented by an equation of the form

A B = C;

but the equation indicates nothing of the nature of the actual operations. Now it may happen, when the operations of two groups of equal order are represented by symbols,

\begin{array}{l} (i) & 1, A, B, C, \dots \\ (ii) & 1, A', B', C', \dots \end{array}

that, to every relation of the form

A B = C

between operations of the ﬁrst group, there corresponds the relation

A' B' = C'

between operations of the second group. In such a case, although the nature of the actual operations in the ﬁrst group may be entirely diﬀerent from the nature of those in the second, the laws according to which the operations of each group combine among themselves are identical. The following series of groups of operations, of order six, will at once illustrate the possibility just mentioned, and will serve as concrete examples to familiarize the reader with the conception of a group of operations.

I. Group of inversions. Let $P$ , $Q$ , $R$ be three circles with a common radical axis and let each pair of them intersect at an angle $\frac{1}{3} π$ . Denote the operations of inversion with respect to $P$ , $Q$ , $R$ by $C$ , $D$ , $E$ ; and denote successive inversions at $P$ , $R$ and at $P$ , $Q$ by $A$ and $B$ . The object of operation may be any point in the plane of the circles, except the two common points in which they intersect. Then it is easy to verify, from the geometrical properties of inversion, that the operations

1, A, B, C, D, E

are all distinct, and that they form a group. For instance,

D E

represents successive inversions at

Q

and

R

. But successive inversions at

Q

and

R

produce the same displacement of points as successive inversions at

P

and

Q

, and therefore

D E = B .

II. Group of rotations. Let $P O P'$ , $Q O Q'$ , $R O R'$ be three concurrent lines in a plane such that each of the angles $P O Q$ and $Q O R$ is $\frac{1}{3} π$ , and let $I O I'$ be a perpendicular to their plane. Denote by $A$ a rotation round $I I'$ through $\frac{2}{3} π$ bringing $P P'$ to $R R'$ ; and by $B$ a rotation round $I I'$ through $\frac{4}{3} π$ bringing $P P'$ to $Q' Q$ . Denote also by $C$ , $D$ , $E$ rotations through two right angles round $P P'$ , $Q Q'$ , $R R'$ . The object of the rotations may be any point or set of points in space. Then it may again be veriﬁed, by simple geometrical considerations, that the operations

1, A, B, C, D, E

are distinct and that they form a group.

III. Group of linear transformations of a single variable. The operation of replacing $x$ by a given function $f (x)$ of itself is sometimes represented by the symbol $(x, f (x))$ . With this notation, if

\begin{array}{l} A & = (x, \frac{1}{1 - x}), & B & = (x, \frac{x - 1}{x}), & C & = (x, \frac{1}{x}), & D & = (x, 1 - x), \\ E & = (x, \frac{x}{x - 1}), & 1 & = (x, x), \end{array}

it may again be veriﬁed without diﬃculty that these six operations form a group.

IV. Group of linear transformations of two variables. With a similar notation, the six operations

\begin{array}{l} A & = (x, \frac{y}{x}; y, \frac{1}{x}), & B & = (x, \frac{1}{y}; y, \frac{x}{y}), & C & = (x, y; y, x), \\ D & = (x, \frac{1}{x}; y, \frac{y}{x}), & E & = (x, \frac{x}{y}; y, \frac{1}{y}), & 1 & = (x, y; x, y) \end{array}

form a group.

V. Group of linear transformations to a prime modulus.

The six operations deﬁned by

\begin{array}{l} A & = (x, x + 1), & B & = (x, x + 2), & C & = (x, 2 x), \\ D & = (x, 2 x + 2), & E & = (x, 2 x + 1), & 1 & = (x, x), \end{array}

where each transformation is taken to modulus $3$ , form a group.

VI. Group of substitutions of $3$ symbols. The six substitutions

1, A = (x y z), B = (x z y), C = x (y z), D = y (z x), E = z (x y)

are the only substitutions that can be formed with three symbols; they must therefore form a group.

VII. Group of substitutions of $6$ symbols. The substitutions

\begin{matrix} 1, A = (x y z) (a b c), B = (x z y) (a c b), C = (x a) (y c) (z b), \\ D = (x b) (y a) (z c), E = (x c) (y b) (z a) \end{matrix}

may be veriﬁed to form a group.

VIII. Group of substitutions of $6$ symbols. The substitutions

\begin{matrix} 1, A = (x a y b z c), B = (x y z) (a b c), C = (x b) (y c) (z a), \\ D = (x z y) (a c b), E = (x c z b y a) \end{matrix}

form a group.

The operations in the ﬁrst seven of these groups, as well as the objects of operation, are quite diﬀerent from one group to another; but it may be shewn that the laws according to which the operations, denoted by the same letters in the diﬀerent groups, combine together are identical for all seven. There is no diﬃculty in verifying that in each instance

A^{3} = 1, C^{2} = 1, B = A^{2}, D = A C = C A^{2}, E = A^{2} C = C A;

and from these relations the complete system, according to which the six operations in each of the seven groups combine together, may be at once constructed. This is given by the following multiplication table, where the left-hand vertical column gives the ﬁrst factor and the top horizontal line the second factor in each product; thus the table is to be read

A 1 = A

A B = 1

A C = D

, and so on.

	1	A	B	C	D	E


1	1	A	B	C	D	E
A	A	B	1	D	E	C
B	B	1	A	E	C	D
C	C	E	D	1	B	A
D	D	C	E	A	1	B
E	E	D	C	B	A	1

But, though the operations of the seventh and eighth groups are of the same nature and though the operands are identical, the laws according to which the six operations combine together are quite distinct for the two groups. Thus, for the last group, it may be shewn that

B = A^{2}, C = A^{3}, D = A^{4}, E = A^{5}, A^{6} = 1,

so that the operations of this group may, in fact, be represented by

1, A, A^{2}, A^{3}, A^{4}, A^{5} .

18. If we pay no attention to the nature of the actual operations and operands, and consider only the number of the former and the laws according to which they combine, the ﬁrst seven groups of the preceding paragraph are identical with each other. From this point of view a group, abstractly considered, is completely deﬁned by its multiplication table; and, conversely, the multiplication table must implicitly contain all properties of the group which are independent of any special mode of representation.

It is of course obvious that this table cannot be arbitrarily constructed. Thus, if

A B = P and B C = Q,

the entry in the table for

P C

must be the same as that for

A Q

. Except in the very simplest cases, the attempt to form a consistent multiplication table, merely by trial, would be most laborious.

The very existence of the table shews that the symbols denoting the diﬀerent operations of the group are not all independent of each other; and since the number of symbols is ﬁnite, it follows that there must exist a set of symbols $S_{1}$ , $S_{2}$ , …, $S_{n}$ no one of which can be expressed in terms of the remainder, while every operation of the group is expressible in terms of the set. Such a set is called a set of fundamental or generating operations of the group. Moreover though no one of the generating operations can be expressed in terms of the remainder, there must be relations of the general form

S_{m}^{a} S_{n}^{b} \dots S_{p}^{c} = 1

among them, as otherwise the group would be of inﬁnite order; and the number of these relations, which are independent of one another, must be ﬁnite. Among them there necessarily occur the relations

S_{1}^{a_{1}} = 1, S_{2}^{a_{2}} = 1, \dots, S_{n}^{a_{n}} = 1

giving the orders of the fundamental operations.

We thus arrive at a virtually new conception of a group; it can be regarded as arising from a ﬁnite number of fundamental operations connected by a ﬁnite number of independent relations. But it is to be noted that there is no reason for supposing that such an origin for a group is unique; indeed, in general, it is not so. Thus there is no diﬃculty in verifying that the group, whose multiplication table is given in § 17, is completely speciﬁed either by the system of relations

A^{3} = 1, C^{2} = 1, {(A C)}^{2} = 1,

or by the system

C^{2} = 1, D^{2} = 1, {(C D)}^{3} = 1 .

In other words, it may be generated by two operations of orders $2$ and $3$ , or by two operations of order $2$ . So also the last group of § 17 is speciﬁed either by

A^{6} = 1,

or by

B^{3} = 1, C^{2} = 1, B C = C B .

19. Deﬁnition. Let $G$ and $G'$ be two groups of equal order. If a correspondence can be established between the operations of $G$ and $G'$ , so that to every operation of $G$ there corresponds a single operation of $G'$ and to every operation of $G'$ there corresponds a single operation of $G$ , while to the product $A B$ of any two operations of $G$ there corresponds the product $A' B'$ of the two corresponding operations of $G$ ’, the groups $G$ and $G'$ are said to be simply isomorphic⁴.

Two simply isomorphic groups are, abstractly considered, identical. In discussing the properties of groups, some deﬁnite mode of representation is, in general, indispensable; and as long as we are dealing with the properties of a group per se, and not with properties which depend on the form of representation, the group may, if convenient, be replaced by any group which is simply isomorphic with it. For the discussion of such properties, it would be most natural to suppose the group given either by its multiplication table or by its fundamental operations and the relations connecting them; and as far as possible we shall follow this course. Unfortunately, however, these purely abstract modes of representing a group are by no means the easiest to deal with. It thus becomes an important question to determine as far as possible what diﬀerent concrete forms of representation any particular group may be capable of; and we shall accordingly end the present chapter with a demonstration of the following general theorem bearing on this question.

20. THEOREM. Every group of ﬁnite order $N$ is capable of representation as a group of substitutions of $N$ symbols⁵.

Let

1, S_{1}, S_{2}, \dots, S_{i}, \dots, S_{N - 1}

be the

N

operations of the group; and form the complete multiplication table

\begin{matrix} 1, & S_{1}, & S_{2}, & \dots, & S_{i}, & \dots, & S_{N - 1}, \\ S_{1}, & S_{1}^{2}, & S_{2} S_{1}, & \dots, & S_{i} S_{1}, & \dots, & S_{N - 1} S_{1}, \\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . \\ S_{i}, & S_{1} S_{i}, & S_{2} S_{i}, & \dots, & S_{i}^{2}, & \dots, & S_{N - 1} S_{i}, \\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . \\ S_{N - 1}, & S_{1} S_{N - 1}, & S_{2} S_{N - 1}, & \dots, & S_{i} S_{N - 1}, & \dots, & S_{N - 1}^{2}, \end{matrix}

that results from multiplying the symbols in the ﬁrst horizontal line by

1

S_{1}

S_{2}

, …,

S_{N - 1}

in order. Each horizontal line in the table so obtained contains the same

N

symbols as the original line, but any given symbol occupies a diﬀerent place in each line; for the supposition that the two symbols

S_{i} S_{p}

and

S_{i} S_{q}

were identical would involve

S_{p} = S_{q},

which is not true.

It thus appears that the ﬁrst line in the table, taken with the $(i + 1)$ th line, deﬁnes a substitution

(\binom{1, S_{1}, S_{2}, \dots, S_{N - 1}}{S_{i}, S_{1} S_{i}, S_{2} S_{i}, \dots, S_{N - 1} S_{i}})

performed on the

N

symbols

1

S_{1}

S_{2}

, …,

S_{N - 1}

; and by taking the ﬁrst line with each of the others, including itself, a set of

N

substitutions performed on these

N

symbols is obtained. Now this set of substitutions forms a group simply isomorphic to the given group. For if

S_{p} S_{i} = S_{q} and S_{q} S_{j} = S_{r},

the substitutions

(\binom{1, S_{1}, S_{2}, \dots, S_{N - 1}}{S_{i}, S_{1} S_{i}, S_{2} S_{i}, \dots, S_{N - 1} S_{i}})

and

(\binom{1, S_{1}, S_{2}, \dots, S_{N - 1}}{S_{j}, S_{1} S_{j}, S_{2} S_{j}, \dots, S_{N - 1} S_{j}}),

successively performed, change

S_{p}

into

S_{r}

But

S_{p} S_{i} S_{j} = S_{r},

and therefore the product of the two substitutions, in the order given, is the substitution

(\binom{1, S_{1}, S_{2}, \dots, S_{N - 1}}{S_{i} S_{j}, S_{1} S_{i} S_{j}, S_{2} S_{i} S_{j}, \dots, S_{N - 1} S_{i} S_{j}}),

or the product of any two of the

N

substitutions is again one of the substitutions of the set. Moreover the above reasoning shews that the result of performing successively the substitutions corresponding to the operations

S_{i}

and

S_{j}

gives the substitution corresponding to the operation

S_{i} S_{j}

. Hence, since the number of substitutions is equal to the number of operations, the given group and the group of substitutions are simply isomorphic.

It may also be shewn that each of the $N$ substitutions is regular (§ 9) in the $N$ symbols. For the substitution

(\binom{1, S_{1}, S_{2}, \dots, S_{N - 1}}{S_{i}, S_{1} S_{i}, S_{2} S_{i}, \dots, S_{N - 1} S_{i}})

changes

S_{r}

into

S_{r} S_{i}

S_{r} S_{i}

into

S_{r} S_{i}^{2}

, and so on. If then

m

is the order of

S_{i}

, the cycle of the substitution which contains

S_{r}

will be

(S_{r}, S_{r} S_{i}, S_{r} S_{i}^{2}, \dots, S_{r} S_{i}^{m - 1}) .

Now $S_{r}$ may be any symbol of the set; hence all the cycles of the substitution must contain the same number, $m$ , of symbols.

The substitution is therefore regular in the $N$ symbols, and this can only be the case if $m$ is a factor of $N$ . It follows at once, as was stated in § 15, that the order of any operation of a group of order $N$ must be equal to or a factor of $N$ .

All the substitutions in this form of representing a group being regular, the group itself is said to be expressed as a regular substitution group.

21. The form, in which it has just been shewn that every group can be represented, is by no means the only form of representation possessing this property. Thus Hurwitz⁶ has shewn that every group can be expressed as a group of rational and reversible transformations which change a suitably chosen algebraic curve (or Riemann’s surface) into itself. We shall see in Chapter XIV that every group can also be expressed as a group of linear transformations of a ﬁnite set of variables to a prime modulus.

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Chapter II.The Deﬁnition of a Group.

Chapter II.
The Deﬁnition of a Group.