A cryptographic system, called PGM, was invented in the late 1970’s by S. Magliveras. PGM is based on the prolific existence of certain kinds of factorization sets, called logarithmic signatures, for finite permutation groups. Logarithmic signatures were initially motivated by C. Sims’ bases and strong generators. Statistical properties of random number generators based on PGM have been investigated in ,  and show PGM to be statistically robust. In this paper we present recent results on the algebraic properties of PGM. PGM is an endomorphic cryptosystem in which the message space is Z|G|, for a given finite permutation group G. We show that the set of PGM transformations TG is not closed under functional composition and hence not a group. This set is 2-transitive on Z|G| if the underlying group G is not hamiltonian. Moreover, if |G| ≠ 2a, then the set of transformations contains an odd permutation. An important consequence of the above results is that the group generated by the set of transformations is nearly always the full symmetric group.