Kurosh subgroup theorem

In the

History and generalizations

After the original 1934 proof of Kurosh, there were many subsequent proofs of the Kurosh subgroup theorem, including proofs of Harold W. Kuhn (1952),

Let ${\displaystyle G=A*B}$ be the free product of groups A and B and let ${\displaystyle H\leq G}$ be a subgroup of G. Then there exist a family ${\displaystyle (A_{i})_{i\in I}}$ of subgroups ${\displaystyle A_{i}\leq A}$, a family ${\displaystyle (B_{j})_{j\in J}}$ of subgroups ${\displaystyle B_{j}\leq B}$, families ${\displaystyle g_{i},i\in I}$ and ${\displaystyle f_{j},j\in J}$ of elements of G, and a subset ${\displaystyle X\subseteq G}$ such that

${\displaystyle H=F(X)*(*_{i\in I}g_{i}A_{i}g_{i}^{-1})*(*_{j\in J}f_{j}B_{j}f_{j}^{-1}).}$

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.^[9] Its formulation is:

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

${\displaystyle H=F(X)*(*_{j\in J}g_{j}H_{j}g_{j}^{-1}),}$

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

Proof using Bass–Serre theory

The Kurosh subgroup theorem easily follows from the basic structural results in Bass–Serre theory, as explained, for example in the book of Cohen (1987):^[8]

Let G = A∗B and consider G as the fundamental group of a graph of groups Y consisting of a single non-loop edge with the vertex groups A and B and with the trivial edge group. Let X be the Bass–Serre universal covering tree for the graph of groups Y. Since H ≤ G also acts on X, consider the quotient graph of groups Z for the action of H on X. The vertex groups of Z are subgroups of G-stabilizers of vertices of X, that is, they are conjugate in G to subgroups of A and B. The edge groups of Z are trivial since the G-stabilizers of edges of X were trivial. By the fundamental theorem of Bass–Serre theory, H is canonically isomorphic to the fundamental group of the graph of groups Z. Since the edge groups of Z are trivial, it follows that H is equal to the free product of the vertex groups of Z and the free group F(X) which is the fundamental group (in the standard topological sense) of the underlying graph Z of Z. This implies the conclusion of the Kurosh subgroup theorem.

Extension

The result extends to the case that G is the

• HNN extension
• Geometric group theory
• Nielsen–Schreier theorem

References