We count the finitely generated subgroups of modular group [Formula: see text]. More precisely, each such subgroup text] can be represented by its Stallings graph text], we consider number vertices to size and Since an index has our results generalize known on enumeration finite give asymptotic equivalents for as well subgroups, free subgroups. also expected value isomorphism type a prove large...

Let G be a group, S a subgroup of G, and F a field of characteristic p. We denote the augmentation ideal of the group algebra FG by ω(G). The Zassenhaus-Jennings-Lazard series of G is defined by Dn(G) = G ∩ (1 + ω(G)). We give a constructive proof of a Theorem of Quillen stating that the graded algebra associated to FG is isomorphic as an algebra to the enveloping algebra of the restricted Lie ...