ABSTRACT GROUPS A group may be defined without reference to the properties of the particular elements of which it is composed. Indeed, these elements need have no mathematical properties other than those defined by the rules of combination. When so defined the group is called an abstract group. An abstract group is defined by E. H. Moore to be a system consisting of a set of elements a, b, c, and a mode of combination, ordinarily called multiplication, which satisfies the following postulates: (I) The product of any two elements a and b of the set, whether a and b denote two distinct elements or the same element, is a unique element c, which is an element of the set. In symbols, ab=c.
A group is said to be finite or infinite according as the number of elements is finite or infinite. By the aid of postulates (3) and (4) it is easy to prove the following theorem: (3a) If ab = ab', then b=b', and if ab = a'b, then a = a'. For it is only necessary to note that left-hand multiplication of both members of the first equation by gives b=b' , and right-hand multiplication of both members of the second by gives a= a'. On the other hand, if the number of elements in the group is finite and (3a) be assumed as a postulate along with (I) and (2), it is easy to prove (3) and (4) as theorems. If the group is infinite it is not possible to prove (3) and (4) without assuming a further postu late: (4a). If two elements a and b are given, a third element c, unique in each case, is determined by any one of the three equa tions, ac = b, ca=b, ab = c. For finite groups the proposition (4a) is a direct consequence of postulates (I), (2), (3) and (4). A finite group may be defined by three postulates, (I), (2) and (3a), and an infinite group by the four postulates, (I), (2), (3a) and (4a) (Weber).
The elements of a closed set do not necessarily form a group since they need satisfy postulate (I) alone. Closed sets lacking the identical element and the inverse elements have been called semi-groups. The positive integral powers of a number which is neither zero nor a root of unity form a semi-group. The number of elements in a finite group is called the order of the group. If there exists within a group G a set which satisfies the group property, i.e., postulate (I), this set is called a subgroup of G. A group for which the product of every two elements is commuta tive is called an Abelian group. The simplest example is the group composed of the powers of a single substitution of prime order. The customary notation for a group whose elements are a, b, c, isG=[a,b,c,].
Examples of Groups.---I. The n-factorial substitutions on n letters form a group of order n! called the symmetric group. The even substitutions of the symmetric group form a sub-group of order n . called the alternating group. The odd substitutions 2 of the symmetric group, or, for that matter, of any substitution group, do not form a group since the product of two odd sub stitutions is even. A substitution group is said to be transitive if it contains at least one substitution which will replace an ar bitrary given letter by another arbitrary given letter, otherwise it is intransitive. Transitivity is not an essential property of a group since the same group may be represented by both transi tive and intransitive substitution groups.
2. The rotations of a regular solid into itself form a group in which the product of two elements is defined to be the rotation equivalent to the two rotations performed in succession. Included among these rotation groups are the dihedral groups which are obtained by rotating regular polygons into themselves, or what amounts to the same thing, rotating double right pyramids with regular bases into themselves. The rotations of a polygon into itself without turning it over form a cyclic group. The groups of the cube and the dodecahedron are the same as those of their respective polar figures, the octahedron and the icosahedron. The polyhedral groups, i.e., the cyclic, dihedral, tetrahedral, octahedral and icosahedral groups, complete the enumeration of the finite rotation groups.
3. The totality of linear fractional transformations of the form z' = IXz+i3 for which the determinant a3-13-y= i form a group for which multiplication is defined by the elimination of z' from two transformations The determinant of the product ST will be the product of the determinants of the two transformations. The identical trans formation is obtained by making a= 8 = I and 13= y = o, and the inverse is found by solving the transformation for z in terms of z'. If the linear fractional transformation has the form where a= Esin e, b = n sin e, c= d= cos and E, 2 2 2 2 are the co-ordinates of a point on a sphere of unit radius with centre at the origin, it represents a rotation through angle 0 about the diameter through the point (E, 0. The points z and z' of the x+iy-plane are the stereographic projections from the north pole of the sphere upon its equatorial plane of the point f") and the point into which (E, ?') is rotated. This formula, known as Cayley's rotation formula, gives all the polyhedral groups, which are the only finite groups that can be represented as groups of linear fractional transformations on one variable. (See F. Klein's Vorlesungen fiber das Ikosaeder.) Closely related to the polyhedral groups are the crystallo graphic groups whose operations are rotations, reflections, or combinations of these. Every so-called symmetry group defines a definite crystal system, and, conversely, every crystal belongs to a symmetry group. (See Hilton's Mathematical Crystallo graphy.) Groups of linear and linear fractional transformations will be discussed in a later section. If a, 0, y, b are rational in tegers subject to the condition ab - fry = 1, an infinite discrete group, known as the modular group, is obtained. The modular group plays a fundamental role in the theory of the elliptic functions.
Isomorphism.-Two groups are said to be simply isomorphic when the following conditions hold: I. A one-to-one correspondence can be established between the elements of the two groups.
From the point of view of abstract groups, two simply iso morphic groups, are identical; or, we may say they differ only in their mode of representation. The tetrahedral group is simply isomorphic with the alternating group on four letters; the octa hedral group is simply isomorphic with the symmetric group on four letters; and the icosahedral group is isomorphic with the alternating group on five letters.
The prime importance of substitution groups lies in the fact that according to a fundamental theorem due to Cayley (some times attributed to Jordan), any finite group of order N is simply isomorphic with a transitive substitution group on N letters. To prove Cayley's theorem, let the elements of the group G be G= [I, 52,53,,sN] If si be any element of the group, the products Sisi; S3Si, , will be the elements of the group in some order. It follows that -/ , SNl Sisii, S2Si, S3Si, . , s:si is a substitution on the letters , Similarly, if s; be a second element of G, there is formed on the N letters a second substitution Si which corresponds to si. It follows that the product SiSi is in the set Si, 52, and, moreover, corresponds to the product sisi in G. The group F = [S1, 52, , SN], simply isomorphic with G, is transitive and the number of elements is equal to the number of letters. Such a group is called regular. One of the important problem in the group theory is that of finding the smallest number of letters by means of which a given group can be represented as a group of substitutions. Such sub stitution groups will not be regular except in special cases. If the groups G and G' are p-fold and q-fold isomorphic re spectively to a group F, then G and G' are said to be (p-q)-fold isomorphic to each other.