By Arthur Gill
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Additional resources for Applied algebra for the computer sciences
Expressed in modern terms this means ~ and fJ are conjugate within the group s:s of all contact transformations. Expressed in the language of Lie and Klein, it meant that the equations defining the transformations of fJ are transformed into those of ~ by the variable change defined by T. As we have seen (Section 3), Lie and Klein called such groups similar and regarded them as essentially the same. The problem of determining all groups of a particular type always meant classifying them up to similarity.
This, in turn, is related to group classification problems. For example, given a system of first-order partial differential equations, consider the contact transformations T that are "admitted" by the system in Lie's sense (as introduced in Section 3) that T takes solutions of the system into solutions. The totality of all such T certainly has the group property. Thus each system has associated to it a group of contact transformations admitted by the system (which might be the trivial group consisting of the identity transformation).
Hermann pointed out that this can be interpreted as reflecting the accidental isomorphism of types A3 and D3 [1976:38]. 4. The Sphere Mapping 31 suggests, it does not appear to have been a significant motivating factor behind the initial consideration of continuous groups by Lie and Klein. 32 Geometrical considerations naturally led Klein and Lie to continuous groups. That is not to say that Jordan's memoir had no impact upon them. Theyand especially Lie - seem to have viewed it as an example of a successful (albeit limited) group classification project.