In this paper we describe new fundamental properties of the law PΓ of the classical gamma process and related properties of the Poisson-Dirichlet measures PD(θ). We prove the quasi-invariance of the measure P_Γ w ith respect to an infinite-dimensional multiplicative group (the fact first disc overed in [4]) and the Markov-Krein identity as corollaries of the formula for the Laplace transform of P_Γ. The quasi-invariance of the measure P _Γ allows us to obtain new quasi-invariance properties of the mea sure PD(θ). The corresponding invariance properties hold for σ-fini te analogues of P_Γ and PD(θ). We also show that the meas ure P_Γ can be considered as a limit of measures corresponding to the α-stable Lévy processes when the parameter α tends to zero. Our approach is based on considering simultaneously the gamma process (especiall y its Laplace transform) and its simplicial part - the Poisson-Dirichlet measures .