History and Generalizations
After the original 1934 proof of Kurosh, there were many subsequent proofs of the Kurosh subgroup theorem, including proofs of Kuhn (1952), Mac Lane (1958) and others. The theorem was also generalized for describing subgroups of amalgamated free products and HNN extensions. Other generalizations include considering subgroups of free pro-finite products and a version of the Kurosh subgroup theorem for topological groups.
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.
Read more about this topic: Kurosh Subgroup Theorem
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