Kurosh Subgroup Theorem - Statement of The Theorem

Statement of The Theorem

Let G = A∗B be the free product of groups A and B and let H ≤ G be a subgroup of G. Then there exist a family (A_i)_{i ∈ I} of subgroups A_i ≤ A, a family (B_j)_{j ∈ J} of subgroups B_j ≤ B, families g_i, i ∈ I and f_j, j ∈ J of elements of G, and a subset X ⊆ G such that

This means that X freely generates a subgroup of G isomorphic to the free group F(X) with free basis X and that, moreover, g_iA_ig_i−1, f_jB_jf_j−1 and X generate H in G as a free product of the above form.

There is a generalization of this to the case of free products with arbitrarily many factors. Its formulation is:

If H is a subgroup of ∗_i∈IG_i = G, then

where X ⊆ G and J is some index set and g_j ∈ G and each H_j is a subgroup of some G_i.