Let (Gi | i ∈ I) be a family of groups, let F be a free group, and let the free product of F and all the Gi.
Let denote the set of all finitely generated subgroups H of G which have the property that, for each g ∈ G and each i ∈ I, By the Kurosh Subgroup Theorem, every element of is a free group. For each free group H, the reduced rank of H, denoted r(H), is defined as To avoid the vacuous case, we make the additional assumption that contains a non-cyclic group, and we define
We are interested in precise bounds for . In the special case where I is empty, Hanna Neumann proved that ∈ [1,2], and conjectured that = 1; fifty years later, this interval has not been reduced.
With the understanding that ∞/(∞ − 2) is 1, we define
Generalizing Hanna Neumann's theorem we prove that , and, moreover, whenever G has 2-torsion. Since is finite, is closed under finite intersections. Generalizing Hanna Neumann's conjecture, we conjecture that whenever G does not have 2-torsion.