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 (Ai)i ∈ I of subgroups Ai ≤ A, a family (Bj)j ∈ J of subgroups Bj ≤ B, families gi, i ∈ I and fj, 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, giAigi−1, fjBjfj−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∈IGi = G, then
where X ⊆ G and J is some index set and gj ∈ G and each Hj is a subgroup of some Gi.
Read more about this topic: Kurosh Subgroup Theorem
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