An infinite group does not necessarily con tain the element 1 nor the inverse elements. Thus all the motions of a point along a line in one direction form an infinite group, but this does not contain the reverse motions nor the case of no motion. The prevailing tendency is, however, to restrict the name group to systems which contain the inverse of their elements, and consequently the element 1.
A part of the elements of a group G, taken by themselves, may form a group H, which is then called a subgroup of G. Thus the powers of any element x of G form a cyclical group H which is either G itself or a subgroup of G. The tetrahedral group has a subgroup of order 4 composed of b,, b,, b, and identity. The order h of a subgroup H of G is always a divisor of the order g of G. If pi, where p is a prime number, is a divisor of g, G has one or more subgroups of order pi, and the total number of these subgroups is of the form kp 1, where k is an integer. If pa is the highest power of p that divides g, kp +1 is also a divisor of g. These theorems of Sylow and Frobenius are of great assistance in the analysis of groups of finite order. Thus a group of order pq, where p and q are prime numbers, has a single sub group of order p; it has also a single subgroup of order q. unless p is of the form 0+1. Thus the order 15=5.3 presents only one case, while the order 21=7.3 presents two. For a further example, the icosahedral group of rotations, which is of order 60, contains subgroups of orders, 2, 4, 3, 5, and also 6 and 10. The 15 lines joining middle points of opposite edges of the icosahedron form five sets of trirectangular axes, each of which sets is converted into itself by a tetrahedral group contained as subgroup in the icosahcdral group. There are no subgroups of orders 15, 20, or 30 present.
Isomorphism and which have the same algebra are called isomor phic. Written in the same symbols, isomorphic groups are by definition identical. But in the practice the isomorphism requires to be detected, being veiled under dissimilarity of notation. Once detected among groups derived perhaps from quite different mathematical fields, iso morphism constitutes the unifying principle al ready mentioned. For example, the tetrahedral group is isomorphic with the group of 12 sub stitutions (rearrangements) which it produces among the four vertices of the tetrahedron; and the icosahedron group is isomorphic with the corresponding group of substitutions of the five trirectangular axis systems mentioned above. These isomorphisms contribute materially to the theory of equations of degrees four and five.
One instance of isomorphism is expressible by a universal formula. Let G be any group, with elements a, b, c, ..., and let t be any ele ment whatever capable of combination with a, b, c, ..., under conditions (1)—(3) • then the elements a' = b'-=t—lbt, c' = I — ... form a group G' and this group G' is isomorphic with G. For if ab= c, for exam ple, then a'bi=t—lat-t = t—labt=t—la e', so that not only a', b', c form a group, but the algebra of this group is identical with that of G. The process of deriving G' from G is called transformation of G by t; G' is called the transform of G by t. All transforms of a group G (by t, s, ....) are isomorphic with C and with each other.
Transformation has a very simple concrete significance. Suppose that G is a group of operations, a, b, c, .... performed on a field of objects A, and that t converts A into a second field of objects B; then t — i.e., t reversed, followed by G, followed by t, produces among the objects B an effect precisely parallel to that produced by G on the corresponding objects A. For example, if A is a plane, G a group of operations in A, t a projection of A on a second plane B. then t — is the projection of the group G on B. Or again, if G is a group of rotations about an axis 'A, and t a rotation which moves A into the position B, then t is a second group of rotations, precisely similar to G, performed about the new axis B. In gen eral, transformation in the present sense is the concomitant, for operations, of transformations in the ordinary sense as affecting objects.
Group Analysis-- If G is any group and H any subgroup of G, all the transforms of H with respect to the elements of G are contained in G. These transforms are called the conju gates of H in G. Thus the subgroups of order 3 of the tetrahedral group are conjugate in that group. A noteworthy general example is that of the subgroups of order pa ( a a maximum) of a group G; these kp + 1 subgroups are always conjugate. The number of conjugates of any subgroup H of G is a divisor of the order of G. In the important case where its conjugates all coincide, H is an invariant sub group of G. Every group G has two invariant subgroups, itself and the identical operation. If it has no other invariant subgroups, G is simple; otherwise G is compound. Thus the four rotations 1, bs of the tetrahedral group form an invariant subgroup, this beIng in fact the only subgroup of order 4, of the tetrahedral group.