LINEAR GROUPS Linear Homogeneous Substitutions.-A generalization of the notion of a letter substitution of far-reaching importance is the linear homogeneous substitution on n independent variables y1, y2, , yn given by the n equations z,=ally1+a,2y2+ +alnyn
+a2nyn zn = anlyl+an2y2+ +annyn n or, briefly, A :
= E
(i= I, 2, , .n);
or, still more briefly, z = A (y).
The matrix of the coefficients
is called the matrix of the substitution and the corresponding determinant is the determinant of the substitution. A substitution is completely determined by its matrix so that the same notation may be used for both.
looked upon as a special chapter in the theory of matrices. The matrix and determinant of a substitution are denoted respectively 1,..
B. We may write (z) = A (y) = A [B(x)] = AB(x) . The resultant is a linear substitution AB=C : zi= L Pik xk, (i= I, , n), k_1 where
= E
jk ; i.e., the element in the ith row and the ;_1 kth column of C is found by taking the sum of the products of the elements of the ith row of A by the corresponding elements of the kth column of B. The identity is that substitution for which
= 1 (i= i, , n), and all other coefficients are zero. If the determinant aX o, the substitution
found by solving the n equations A for
Y2, ,
is the inverse of A.
jth column of C, it follows that the determinant of the product of two substitutions is the product of their determinants. The associative law for multiplication holds. It follows that the to tality of linear homogeneous substitutions with determinants different from zero form a group, which is in the general case an infinite group. However, by placing proper restrictions on the coefficients
it is possible to find finite groups of linear homogeneous substitutions known by the briefer designation linear groups.
sets of variables
, yn and
,
be related by a linear transformation S :
,
(i= I, . , n, lsiklX0) and if the y's be subjected to a transformation A : yi'
2-1 the x's will undergo a linear transformation A' =
To prove the theorem, let (y') = S(x'), (y') = A (y). Then x' =
=
(y) =
.
A' =
is called the transform of A with respect to S.
of substitutions on n letters (and consequently every finite group) is simply isomorphic with a linear group on n variables. For, let x1, x2, , xn and t = El, E2, , En S= El, E2, , En 'it, n2, , nn be two letter substitutions where Si and ni are permutations of the n letters xi. If s and t be made to correspond to the linear homogeneous substitutions stitution is of finite order depends upon the form to which its matrix may be transformed. To this end it is convenient to make use of the so-called characteristic equation of the sub stitution. This equation may be obtained as follows: If the variables x,,
,
be looked upon as homogeneous point coordinates in space of n-- i dimensions, a point is un changed if its co-ordinates be transformed by a linear substitu tion A into Xx,,
,
It follows that X must satisfy the equation
X a12 aln
X ... a2n = o and ant . ann
or, briefly 4)(X) =o. The same equation is found if we
that E
is a linear form which is changed into a multiple of i-1 itself by the substitution A, for then
= Xcixi The result of eliminating the c's from these n equations is 4)(X) = o.
characteristic equation of A. The determinant 4)(X) is the characteristic determinant and the matrix A XI, where I is the unit matrix, formed by subtracting A from the elements in the principal diagonal of the matrix A, is the characteristic matrix of A. The characteristic determinants and characteristic equations of A and A' =
are identical. The equation 4)(X) = o has no zero root since the absolute term is The sum of the roots, all+a22+a33+ +ann, is called the characteristic of the substitution.
One fundamental property, at once apparent, of a substitu tion of finite order is that the roots of the characteristic equation 4)(X) =o must be roots of unity, for if the linear form
i=1 is changed into X times itself and if A' = I, then must Xm= 1. Another fundamental property, not so easily demonstrated, is that if A is of finite order there is a substitution S such that the transform
is a multiplication where w,,
,
are roots of unity called the multipliers of A. Since the characteristic equation of a substitution and its transform are the same it follows that w,,
,
are the roots of 4)(X) = o. If the group is Abelian, all its elements can be transformed simultaneously into multiplications. The first pub lication of the classical theorem that every substitution of finite order can be transformed into a multiplication was given by Jordan in 1877. Subsequent proofs were given by Lipschitz, Kronecker, Weyr, Moore, Maschke, and Rost.
Every linear group G contains at least one similarity substitution. The totality of these form an in variant subgroup H. From the viewpoint of geometry, every substitution in H leaves the point
,
unchanged. If G contains another substitution, the product
forms a class of substitutions all of which transform
,
into the same new point and all of which transform three given collinear points into the same three new collinear points. The class of substitutions
or any given element of the class, which may be called a representative of the class, is called a collineation. It may happen even when G is an infinite group, that the num ber of collineations is finite. If I, A2,
, A j represent the classes of the group G = II + II A
- + HA ,, the group G/H is called a Collineation Group.
y = A (x) we obtain a linear fractional substitution , ailz1+ailz2+ +ain-1Zn-1--ain A :
= , (i =
... n)
--ann--lzn-H-ann member, then replacing y by wi and x' by zi. The linear frac Yn xn tional group is simply isomorphic with the corresponding col lineation group.
linear group on the n variables
x2, ,
is called reducible if it is possible to find
linear functions E2, ,
A problem of chief interest in the theory of linear groups is the determination of those irreducible linear groups which furnish representations for given abstract groups since from the ir reducible representations any representation can be derived. This problem was most successfully studied by Frobenius in a series of memoirs published in the Berliner Sitzungsberichte between 1896 and 1903, in which he originated and developed the theory of group characteristics. Frobenius' theory has been simplified and extended by Schur, Burnside, Dickson, and Blichfeldt.
An abstract group
S2, S3, , SN] is said to be represented by a group of linear substitutions r= [S1= I, 52, S3, SN'] when to every element Si of G there corresponds a linear substitu tion
with square matrix A.., of order m such that A
and, further, that not every
is singular. Under these assump tions it follows that no A.8., is singular. If the matrices A
are not all distinct, the group F is multiply isomorphic with G. Two representations r= [Si = I,
. ,
] and I" =
= I, 52', , SN'] are equivalent if there exists a linear substitution t on the m symbols with matrix T such that
=
for every k, otherwise the representations are distinct.
x be any quantity, the scalar product Ax is formed by multiplying each element of the matrix A by x. If, now,
,
be n independent variables, we may form the matric sum A
This new matrix A formed by adding all the elements in the ith row and jth column of each constituent matrix to form the element in the ith row and jth column of A is called the group matrix corresponding to the representation of the group G by the linear group F. The elements of the group matrix will be homogeneous linear forms in
,
e.g., the cyclic group on three elements
I,
(123) ,
(13 2) is isomorphic with the group of matrices The
elements of the group matrix are linear forms in the n independent variables x. In case A is a reducible group matrix, it is equivalent to a matrix where Ai are irreducible group matrices of orders
,
called the irreducible components of A.
If a reducible group matrix A be equivalent to a second matrix A' of the same type with irreducible components A1',
, A/ , the components of A and A' will be equivalent in some order. It follows that the number of distinct components is invariant for a given group matrix. The determinant of an irreducible matrix is an irreducible function of the variables x. Moreover, the necessary and sufficient condition that two irreducible group matrices shall be equivalent is that their determinants shall be identical.
When a group G given as a regular letter substitution group is represented by a linear group I' having the form x
x2=xt3,
where a, /3, l' are the subscripts I, 2, 3 in some order the group matrix A of I' is said to be regular. When the group matrix A is transformed into its completely reduced form, it will be found that each component
,
is repeated exactly
times, where
is the order of the component matrix, and that the number of distinct components is equal to the num ber of conjugate sets of elements in the group. It follows that the number r of non-equivalent transitive linear groups into which a regular substitution group breaks up is equal to the total number of sets of conjugate elements in the group.
an abstract group G be represented by a group of linear transformations I' =
= I s2, s3, s '] with matrices
,
to each matrix will correspond a characteristic equation whose roots, which are the multipliers of A, are roots of unity. The sum of these multipliers will be the sum of the coefficients in the principal diagonal of the matrix. This sum has already been defined to be the characteristic of the linear transformation s and is denoted by Since the characteristic equations of a matrix and its transform are identical, there are exactly r distinct characteristics of any irreducible representation ri of G where r is the number of con jugate sets in F. These r characteristics, usually denoted by XI, X
X r are called a set of characteristics or a character of
For a given group G there are exactly r sets of characteristics corresponding to the r irreducible representations of G.
The far-reaching character of the theory of group character istics is indicated by the theorem which asserts that the necessary and sufficient condition that two representations of a group of finite order by means of linear substitutions shall be equivalent, is that the characteristics of each set of conjugates shall be the same in both representations. Not only does the theory throw important light upon the theory of linear groups themselves but its application has led to the discovery of important theorems concerning abstract groups not otherwise proven. One of the most important of these is Burnside's proof that every group, whose order is of the form
(p and q prime), is solvable.
Group Invariants.-In the early part of this article, it was noted that to the various groups of letter substitutions correspond functions of the letters which are unchanged by every element of the group. If
,
be the roots of an equation, every symmetric function S is unchanged by all the substitutions of the symmetric group. Every function having the form
x2, ,
l LS2(x1, X2, ,
where A is the dis criminant of the x's, is unchanged by the substitutions of the alternating group. In general, if under the substitutions of a given group G a function 43 takes p values 00, 01, 02, , (PP-1, any
function of (ko, 4i, , Op-1 will be unchanged by every substitution of G. (See THEORY OF EQUATIONS.) Similarly, if the independent variables
,
operated on by a group G of linear substitutions be changed into the n linear functions
,
any rational function
, x) is called an invariant of the group if
x2(*), (+) =F x x x > . , n \ 1, 2, , n), for every element s of G. Clearly, as in the case of letter sub stitutions, if
, x) be any rational function, not identically zero, any symmetric function of the N functions
.. ,
will be an invariant of the group. Since the n variables
. , x are independent, any n+ i in variants must be connected by an algebraic equation. That n algebraically independent invariants actually exist may be proven easily. Given n algebraically independent invariants, it is always possible to find an (n+ I)st invariant such that every other invariant can be expressed rationally in terms of the n+ I. A further important theorem is that if F be an invariant of an irreducible group it is not possible by means of any linear sub stitution to express F as a function of fewer than n variables. No small part of the value of the group theory to other branches of mathematics depends upon the existence of invariants with respect to certain groups which thereby furnish a means of clas sification of functions.
Linear Substitutions Whose Coefficients Are Marks of a Galois Field.-So far it has been assumed that coefficients of the substitutions of the linear group were ordinary algebraic num bers. Many results of extraordinary elegance and simplicity are obtained by considering these coefficients as numbers or marks in a Galois field. A field may be defined as a set of elements or marks
, such that the rational operations of algebra may be performed on the numbers of the field and the results are again numbers of the field. A Galois field, for which the cus tomary abbreviation is GF [ p"], consists of p" marks where p is a rational prime and n is a positive rational integer.
lb, t,,...,E,, represent a letter or symbol and the E's be allowed to run through the marks of a Galois field, the symbol represents pnm letters. The linear homogeneous substitution A : '_ EI, 2, , m), laiji = 0,
where
are marks of GF [p"], may be shown to be an analytic representation of a substitution on the
letters. The totality of such substitutions must then form a finite group called the general linear homogeneous group denoted by the symbol GLH [m, pn]. The same results may be reached by defining the group as a group of linear homogeneous substitutions on m arbitrary in dependent variables 1, E2, , Sn It follows that the trans formation group whose matrices are formed from marks of a Galois field may be represented as a group of letter substitutions on
letters.
GLH [m, p"] is N=
- I )
p2n) . . . (pnr - pn (m-1)) .
GLH(m, pn) the linear fractional group LF(m, pn), on m- i non-homogeneous variables, is formed as in the section on Collineation Groups. The order of the LF(m, p") is where d is the greatest common divisor of m and p" - I. This group has the extraordinary property that for every case except two, viz., m - 2,
= 2, and m=
p" = 3, it is simple. For a comprehensive development of these ideas the reader is referred to Dickson's Linear Groups with an Exposition of the Galois Field Theory.
