William Burnside

“ The totality of rational numbers, including zero, forms a set of objects which are interchanged among themselves by all operations of the group.
If x1 and x2 are any pair of distinct rational numbers, and y1 and y2 any other pair, there is just one operation of the group which changes x1 and x2 into y1 and y2 respectively. For the equations y1 = ax1 + b, y1 = ax2 + b determine a and b uniquely. The group is therefore doubly transitive in respect of the set of rational numbers. ”

“ In accordance with the general definitions already given, a permutation-group is called transitive or intransitive according as it does or does not contain permutations changing any one of the symbols into any other. It is called imprimitive or primitive according as the symbols can or cannot be arranged in sets, such that every permutation of the group changes the symbols of any one set either among themselves or into the symbols of another set. When a group is imprimitive the number of symbols in each set must clearly be the same. ”

“ In general, if the equation is given arbitrarily, the group will be the symmetric group. The necessary and sufficient condition that the equation may be soluble by radicals is that its group should be a soluble group. When the coefficients in an equation are rational integers, the determination of its group may be made by a finite number of processes each of which involves only rational arithmetical operations. These processes consist in forming resolvents of the equation corresponding to each distinct type of subgroup of the symmetric group whose degree is that of the equation. ”

“ For groups of point-transformations in more than two variables there will be more than one set of differential invariants. For instance, with three variables, one may be regarded as independent and the other two as functions of it, or two as independent and the remaining one as a function. Corresponding to these two points of view, the differential invariants for a curve or for a surface will arise.
If a differential invariant of a continuous group of the plane be equated to zero, the resulting differential equation remains unaltered when the variables undergo any transformation of the group. ”

“ The independent generating operations of a group may be subject to certain relations connecting them, but these must be such that it is impossible by combining them to obtain a relation expressing one operation in terms of the others. ”

“ The determination of a particular operation of a given continuous group depends on assigning special values to each one of a set of parameters which are capable of continuous variation. The first distinction regards the number of these parameters. If this number is finite, the group is called a finite continuous group; if infinite, it is called an infinite continuous group. In the latter case arbitrary functions must appear in the equations defining the operations of the group when these are reduced to an analytical form. ”

“ Such a series is called a composition-series of G. In general it is not unique, since a group may have two or more maximum self-conjugate subgroups. A composition-series of a group, however it may be chosen, has the property that the number of terms of which it consists is always the same, while the factor-groups G/G1, G1/G2, . . ., Gn differ only in the sequence in which they occur. It should be noticed that though a group defines uniquely the set of factor-groups that occur in its composition-series, the set of factor-groups do not conversely in general define a single type of group. ”

“ The determination of the finite equations of a cyclical group, when the infinitesimal operation which generates it is given, will always depend on the integration of a set of simultaneous ordinary differential equations. As a very simple example we may consider the case in which the infinitesimal operation is given by X = x2∂/∂x, so that there is only a single variable. ”

“ This subgroup is known as the derived group, and Lie has called a group perfect when it coincides with its derived group. A simple group, since it contains no self-conjugate subgroup distinct from itself, is necessarily a perfect group.
If G is a given continuous group, G1 the derived group of G, G2 that of G1, and so on, the series of groups G, G1, G2, ... will terminate either with the identical operation or with a perfect group; for the order of Gs+1 is less than that of Gs unless Gs is a perfect group. ”

“ It is to be noticed that the question whether an operation is a definite operation or no may depend on the range of the objects on which it operates. For example, the operations of squaring and extracting the square root are definite inverse operations if the objects are restricted to be real positive numbers, but not otherwise. ”

“ By including a sufficient number of these variables the group must be intransitive in them, and must therefore have one or more invariants. Such invariants are known as differential invariants of the original group, being necessarily functions of the differential coefficients of the original variables. For groups of the plane it may be shown that not more than two of these differential invariants are independent, all others being formed from these by algebraical processes and differentiation. ”

“ The most important of the applications of continuous groups are to the theory of systems of differential equations, both ordinary and partial; in fact, Lie states that it was with a view to systematizing and advancing the general theory of differential equations that he was led to the development of the theory of continuous groups. ”

“ Then, of the operations that can be formed from S1, S2, . . ., Sn, the set ΣH, and no others, reduce to the same operation Σ when the conditions T1 = 1, T2 = 1, . . . are satisfied by the generating operations. Hence the group which is generated by the given operations, when subjected to the conditions just written, is simply isomorphic with the factor-group G/H. Moreover, this is obviously true even when the conditions are such that the generating operations are no longer independent. ”

“ If each angle is a submultiple of two right angles or zero, the construction is still effective to give a set of non-overlapping regions, which represent graphically the group which arises from pairs of inversions in the sides of the polygon. ”

“ If G is an integrable group of order r, the infinitesimal operations X1, X2, ..., Xr which generate the group may be chosen so that X1, X2, ..., Xr1, (r1 < r) generate the first derived group, X1, X2, ..., Xr2, (r2 < r1) the second derived group, and so on. When they are so chosen the constants cijs are clearly such that if rp < i < rp+1, rq < j < rq+1, p > q, then cijs vanishes unless s < rp+1. ”