In the mathematical field of group theory, the Kurosh subgroup theorem describes the algebraic structure of subgroups of free products of groups.
The theorem was obtained by Alexander Kurosh, a Russian mathematician, in 1934.
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.
There is a generalization of this to the case of free products with arbitrarily many factors.
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.
By the fundamental theorem of Bass–Serre theory, H is canonically isomorphic to the fundamental group of the graph of groups 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.