Kurosh subgroup theorem

After the original 1934 proof of Kurosh, there were many subsequent proofs of the Kurosh subgroup theorem, including proofs of Harold W. Kuhn (1952),[2] Saunders Mac Lane (1958)[3] and others. The theorem was also generalized for describing subgroups of amalgamated free products and HNN extensions.[4][5] Other generalizations include considering subgroups of free pro-finite products[6] and a version of the Kurosh subgroup theorem for topological groups.[7]

In modern terms, the Kurosh subgroup theorem is a straightforward corollary of the basic structural results of Bass–Serre theory about groups acting on trees.[8]

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.

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.

The result extends to the case that G is the amalgamated product along a common subgroup C, under the condition that H meets every conjugate of C only in the identity element.[10]

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