x2 + y2 + z2 − 1 = 0 and x2 + y2 − z2 − 1 = 0
respectively into themselves. Since one of these quadrics is changed into the other by the imaginary transformation
x′ = x, y′ = y, z′ = z√ (−1),
the general continuous groups which transform the two quadrics respectively into themselves are simply isomorphic. This is not, however, the case for the real continuous groups. In fact, the second quadric has two real sets of generators; and therefore the real group which transforms it into itself has two self-conjugate subgroups, either of which leaves unchanged each of one set of generators. The first quadric having imaginary generators, no such self-conjugate subgroups can exist for the real group which transforms it into itself; and this real group is in fact simple.
Among the groups isomorphic with a given continuous group there The adjunct group. is one of special importance which is known as the adjunct group. This is a homogeneous linear group in a number of variables equal to the order of the group, whose infinitesimal operations are defined by the relations
| Xi = Σi, s cijs xi | ∂ | , (j = 1, 2, ..., r), |
| ∂xs |
where cijs are the often-used constants, which give the combinants of the infinitesimal operations in terms of the infinitesimal operations themselves.
That the r infinitesimal operations thus defined actually generate a group isomorphic with the given group is verified by forming their combinants. It is thus found that (XpXq) = Σs cpqsXs. The X’s, however, are not necessarily linearly independent. In fact, the sufficient condition that Σj ajXj should be identically zero is that Σj ajcijs should vanish for all values of i and s. Hence if the equations Σj ajcijs = 0 for all values of i and s have r′ linearly independent solutions, only r − r′ of the X’s are linearly independent, and the isomorphism of the two groups is multiple. If Y1, Y2, ..., Yr are the infinitesimal operations of the given group, the equations
Σj ajcijs = 0, (s, i = 1, 2, ..., r)
express the condition that the operations of the cyclical group generated by Σj ajYi should be permutable with every operation of the group; in other words, that they should be self-conjugate operations. In the case supposed, therefore, the given group contains a subgroup of order r′ each of whose operations is self-conjugate. The adjunct group of a given group will therefore be simply isomorphic with the group, unless the latter contains self-conjugate operations; and when this is the case the order of the adjunct will be less than that of the given group by the order of the subgroup formed of the self-conjugate operations.