GROUPS, Theory of. Everywhere in mathematics are encountered systems of opera tions, possessing definite laws of combination. Thus, two geometric motions compound into a single motion, two algebraic transformations into a single transformation, under laws as definite as the primordial 2 X 2 of arithmetic but otherwise capable of infinite variety of sim plicity and intricacy. Consider, for example, the 12 rotations of a regular tetrahedron into itself. Any two of these rotations compound into a third one among them, easily identified on a model. By a simple convention, these vari ous combinations can registered in algebraic form. The several rotations may be designated by the marks, a, b, c, ; the symbol ab may indicate that a is followed by b, and at the same time designate their resultant effect. This resultant ab is called the product of a and b in the order written; it is itself one of the 12 rotations, say c, and we write a b = c. It is an instructive exercise to tabulate the products of two or more of the 12 rotations, identifying each product with one of the 12 original rota tions. It is possible to express all the 12 rota tions as products of two of them, say of the rotation a through 120 degrees about an axis through one of the four vertices of the tetrahedron and the rotation b through 180 degrees about an axis joining the middle points of two opposite edges. It may be noted that the products ab and ba are here not the same rotation: a and b are not commutative as in ordinary algebra. On the other hand aa, which is a rotation through 240 degrees about the axis of a, is conveniently denoted by as and b', both of which restore every point to its initial position, may appropriately be equated to 1 (identity), which is included among the 12 rotations. The three rotations b., b, about the (trirectangular) axes joining the middle points of opposite edges of the tetrahedron will be found to be commutative; in fact b2bi-= b,b, = b,, bible= bath= b,, bath = b2; = 1).
The tetrahedral rotations furnished a simple instance of an algebra of operations. Any sys tem of operations possesses such an algebra, of greater or less extent. And, as many different systems of operations, taken from widely sepa rated mathematical fields, often present one and the same algebra, these algebras are worthy of study by themselves, as generalizing and unify ing instruments. Since each algebra is com
pletely defined by the laws of combination of the symbols a, b, c,...., we may abstract the idea of operation entirely and deal with the pure algebra. Thi., position having been reached, it is inevitable to the mathematical mind to reverse the order of thought and to devise algebras a priori, leaving their concrete interpretation for secondary consideration. In constructing such algeb.as, choice among the infinite possibilities will be dominated by the two principles of generality and usefulness. The two qualities are combined in high degree in the algebra of groups.
Definition of Group.-- A system of symbols, or elements, a, b, c, . . . . (finite or infinite in number), conceived as capable of multiplication with each other, is said to form a group if the following conditions are fulfilled: (1) The product of any two elements of i the system is a third element of the system.
(2) The multiplication is associative. (ab)c=a(bc), (but not necessarily commu tative: ab and ba need not be equal).
- (3) Equalities ab=a- b' or ab=a'b require b=b' or a=a', respectively.
The order of a group is the number n of its elements. A group is briefly called finite or infinite, according as its order is finite or infinite.
The defining conditions (1)—(3), classic in their simplicity, possess a most extraordinary fecundity. From them alone proceed, by pure logical deduction, the vast and intricate systems which make up the algebra of groups.
As a primary deduction it may be noted that every finite group G contains one and only one element, identity (denoted by 1), such that for every element x of G lx=xl=x. A proper power xm of any element x of G is equal to this element 1; the lowest exponent m for which this is true is called the order of x; every power of x is equal to one of the m powers, . . , 1. The inverse of x is defined by = 1= xm whence by (3) x—i=xia—'; then etc. The analogy to ordinary algebra (of mth roots of unity) is here perfect. These elementary prin ciples may be illustrated by reference to the tetrahedral group G of order 12 above.