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As a corollary of this theorem, we have8.3 Transformation groups 97Theorem 8.5 (Egorov). Every G 4 contains a G 3 (locally); (see Petrov(1966), p. 180).Proof. If A E ̸= 0, the Jacobi identities yield A B C B CD = 0, showingthat the derived algebra is three-dimensional (at most). If A E = 0 theforms (8.17), (8.18) show clearly that the derived algebra is again at mostthree-dimensional. In all cases the derived algebra (together, if necessary,with enough linearly independent vectors to make the dimension three)generates a G 3 .A slightly different proof was found by Kantowski (see Collins (1977a)).Patera and Winternitz (1977) have explicitly calculated all subgroups G 2and G 3 of the real G 4 .Another result due to Egorov (see Petrov (1966), p. 180) isTheorem 8.6 Every G 5 contains a subgroup G 4 .8.3 Transformation groupsLet M be a differentiable (analytic) manifold and G a Lie group of rparameters. An action of G on M is an (analytic) map µ : G ×M →M;(q, p) → τ q p. Each element q of G is associated with a transformationτ q : M→M. It is assumed that the identity q 0 of G is associated withthe identity map I : p → p of M, and thatτ q τ q ′p = τ qq ′p (8.19)so that the transformations τ q form a group isomorphic with G. The groupis said to be effective (and the parameters essential) ifτ q = I impliesq = q 0 ; only such groups need be considered.The orbit (or trajectory, orminimum invariant variety) ofG through agiven p in M is defined to be O p = {p ′ : p ′ ∈Mand p ′ = τ q p for some q ∈G}. It is a submanifold of M. The group G is said to be transitive on itsorbits, and to be either transitive on M (when O p = M) orintransitive(O p ̸=M). It is simply-transitive on an orbit if τ q p = τ q ′p implies q = q ′ ;otherwise it is multiply-transitive. A group may be simply-transitive ongeneral orbits but multiply-transitive on some special orbit(s). The set ofq in G such that τ q p = p forms a subgroup of G called the stability groupS(p) ofp. Ifp ′ ∈O p , so that there is a q in G such that τ q p = p ′ , and ifq ′ ∈ S(p), then τ q τ q ′τ q −1p ′ = p ′ and hence qq ′ q −1 ∈ S(p ′ ). Thus S(p) andS(p ′ ) are conjugate subgroups of G, and have the same dimension, s say;for brevity, one often refers to the stability subgroup S s of an orbit.For each orbit, a map µ p : G →O p ; q → τ q p can be defined. The map(µ p ) ∗ then maps the right-invariant vector fields on G to vector fields