Kurosh subgroup theorem

In the mathematical field of group theory, the Kurosh subgroup theorem descibes the algebraic structure of subgroups of free products of groups. The theorem was obtained by a Russian mathematician Alexander Kurosh in 1934. [A. G. Kurosh, "Die Untergruppen der freien Produkte von beliebigen Grouppen." Mathematische Annalen, vol. 109 (1934), pp. 647-660.] Informally, the theorem says that every subgroup of a free product is itself a free product of a free group and of groups conjugate to the subgroups of the factors of the original free product.

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) [H. W. Kuhn. "Subgroup theorems for groups presented by generators and relations." Annals of Mathematics (2), vol. 56, (1952), pp. 22-46] , Mac Lane (1958) [S. Mac Lane."A proof of the subgroup theorem for free products." Mathematika, vol. 5 (1958), pp. 13-19] and others. The theorem was also generalized for describing subgroups of amalgamated free products and HNN extensions. [A. Karrass, and D. Solitar. "The subgroups of a free product of two groups with an amalgamated subgroup."
Transactions of the American Mathematical Society, vol. 150 (1970), pp. 227-255.] [A. Karrass, and D. Solitar. "Subgroups of HNN groups and groups with one defining relation". Canadian Journal of Mathematics, vol. 23 (1971), pp. 627-643.] Other generalizations include considering subgroups of free pro-finite products [P. A. Zalesskii, "Open subgroups of free profinite products over a profinite space of indices." (in Russian) Dokl. Akad. Nauk BSSR, vol. 34 (1990), no. 1, pp. 17-20.] and a version of the Kurosh subgroup theorem for topological groups. [P. Nickolas. "A Kurosh subgroup theorem for topological groups." Proceedings of the London Mathematical Society (3), vol. 42 (1981), no. 3, pp. 461-477 ]

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.Daniel Cohen. "Combinatorial group theory: a topological approach." London Mathematical Society Student Texts, 14. Cambridge University Press, Cambridge, 1989. ISBN: 0-521-34133-7; 0-521-34936-2]

tatement 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 ("A"_"i")_"i"∈"I" of subgroups "A"_"i" ≤"A", a family ("B"_"j")_"j"∈"J" of subgroups "B"_"j" ≤"B", families "g"_"i","i"∈"I" and "f"_"j","j"∈"J" of elements of "G", and a subset "X"⊆"G" such that

:$H=(*\_\{iin\; I\}g\_i\; A\_ig\_i^\{-1\})*\; (*\_\{jin\; J\}f\_jB\_jf\_j^\{-1\})*\; F(X).$

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"_"i""A"_"i""g"_"i"^-1, "f"_"j""B"_"j""f"_"j"^-1 and "X" generate "H" in "G" as a free product of the above form.

There is a similar statement describing subgroups of free products with arbitrarily many factors.

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):

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.

ee also

*Free product
*HNN extension
*Bass-Serre theory
*Graph of groups
*Geometric group theory

References