A maximum invariant subgroup H of G is not contained in any larger invariant subgroup of G. A principal series of composition of G consists of G and a series of subgroups, H, I, ..., 1 of G, each of which is a maximum invariant subgroup of the preceding 'one. The ratios of the orders g, h, i, j, .. . 1 of these subgroups is a principal series of factors of composition of G. Apart from their order of succession, these factors of composition remain the same for every principal series of composi tion of G. They play an important part in the theory of algebraic equations.
Every compound group G is reducible to a sequence of simple groups, whose orders are the factors of composition of G. Only simple groups present new problems. The chief problem of the pure theory of groups is there fore the determination of all simple groups. This problem awaits solution. All groups of prime order are simple. Simple groups of com posite order are of rare occurrence, the only cases below order 2000 being one group for each of the orders 60, 168, 360, 504, 660, 1092. The number of different prime factors in the order of a simple group not of prime order is at least three, and the total number of prime factors is at least six (orders 60, 168, 660, 1092 being the only exceptions). No simple groups of odd or der have as yet been found. Several series of orders of are known, for example, iin!(n >4), %pa (pa-- 1), (p etc.
A group whose elements are commutative is called an abelian group. Every subgroup and every element of an abelian group is invariant. The factors of composition are here the prime factors of the order.
Example of Group Construction.- Let a and b be two elements of prime orders p and q (aP=1-= bet) and subject to the further con dition We find successively b-'a'b . „ b - = aji; b - = b (b - 'aib) b b - luiib = ado, b - = , b - . . , lai1)=ai=aiiii. Hence is I (mod. p). If p -1 is not divisible by q, i must be 1, b ab - a, ab =.' ba, that is, a and b are commutative, and their various products ab form an abelian group of order pq. (This group consists of the powers of one element, say ab): But if q divides p-1, the congruence 1 has roots i different from.l. Any one of these roots i having been chosen, the conditions a =1>q, b- kib=ai are consistent and lead again to a group of order pq composed of the distinct products alebi; the last group is non-abelian.
For g=2, the second group presents the multiplication table Substitution Groups.- The permutations or substitutions of n given letters xi, xi, .... xn form a group (the symmetric group) of order oil. The order of any group of substitutions of n letters is a divisor of n 1. An individual substitution is written in cycles thus (xi x, (x, x. xe x,), or sim ply (1 2 3 ) (4 5 6 7), signifies that xi x, xs are to be replaced by x, x, x,, and er,x,x, x, by x,x,x,x4. Every finite group is express ible as (isomorphic with) a substitution group. Thus in the case of the group of order 6 above, if we denote the elements 1, a, a', b, ab, a'b for convenience by x,, xi, ...., .r., the six lines of the table are obtained from the first line by the six substitutions 1, (123) (456), (132) (465), (14) (26) (35), (15) (24) (36), (16)(25)(34), which form a substitution group isomorphic with the original group.
. Those substitutions of n letters x,, x,, .... x,, which leave a given function of ii, xl, xs, unchanged in form, form a group. Thus the function 01- xix, x,x, is unchanged by the eight substitutions G, • 1, (12), (34), (12)(34), (13)(24), (14)(23), (1324), (1423). The sub stitution t : (23) converts 01 into :car, + and transforms the group G, of sb, into the group 1, (13), (24), (13)(24), (12)(34), (14)(23), (1234), (1432) of Interesting examples of substitution groups may also be obtained by determining those sub stitutions of n letters xi, X2, • • • Xn Which trans form the substitution (12...n) into its powers. If n is a prime number, the order of this (meta cyclic) group is n(n-1).
For further discussion of substitution groups see the article GALOIS THEORY OF EQUATIONS.
Groups of Linear Transformations.-All the linear transformations of a complex vari able Is, = (az + /3) / (y,is +3), for which ad - Py * 0, form a group. For two of them in succession evidently amount to a third linear transformation. Thus and give TS: (1-e). The group of all linear trans formations of z is both infinite and continuous. If a, /3 y, (5 are restricted to integral values, the resulting group is still infinite but discontin uous. The modular group is subject to the still further condition ad - /3y =1; this is the group connecting the values of the ratio e' of the two periods of the elliptic integral u = f (40- g,) -ids.