Kurosh subgroup theorem

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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.[1] Informally, the theorem says that every subgroup of a free product is itself a free product of a free group and of its intersections with the conjugates of the factors of the original free product.

History and generalizations[edit]

After the original 1934 proof of Kurosh, there were many subsequent proofs of the Kurosh subgroup theorem, including proofs of Kuhn (1952),[2] 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]

Statement of the theorem[edit]

Let G = AB be the free product of groups A and B and let HG be a subgroup of G. Then there exist a family (Ai)iI of subgroups AiA, a family (Bj)jJ of subgroups BjB, families gi, iI and fj, jJ of elements of G, and a subset XG such that

H=F(X)*(*_{i\in I} g_i A_ig_i^{-1})* (*_{j\in J} f_jB_jf_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, giAigi−1, fjBjfj−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∈IGi = G, then

H=F(X)*(*_{j\in J} g_jH_jg_j^{-1}),

where XG and J is some index set and gjG and each Hj is a subgroup of some Gi.

Proof using Bass–Serre theory[edit]

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 = AB 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 HG 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[edit]

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

See also[edit]

Notes[edit]

  1. ^ A. G. Kurosh, Die Untergruppen der freien Produkte von beliebigen Gruppen. Mathematische Annalen, vol. 109 (1934), pp. 647–660.
  2. ^ H. W. Kuhn. Subgroup theorems for groups presented by generators and relations. Annals of Mathematics (2), vol. 56, (1952), pp. 22–46
  3. ^ S. Mac Lane. A proof of the subgroup theorem for free products. Mathematika, vol. 5 (1958), pp. 13–19
  4. ^ 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.
  5. ^ 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.
  6. ^ Zalesskii, Pavel Aleksandrovich (1990). "[Open subgroups of free profinite products over a profinite space of indices]". Doklady Akademii Nauk SSSR (in Russian) 34 (1): 17–20. 
  7. ^ P. Nickolas. A Kurosh subgroup theorem for topological groups. Proceedings of the London Mathematical Society (3), vol. 42 (1981), no. 3, pp. 461–477
  8. ^ a b 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
  9. ^ William S. Massey, Algebraic topology: an introduction. Graduate Texts in Mathematics, Springer-Verlag, New York, 1977, ISBN 0-387-90271-6; pp. 218–225
  10. ^ Serre, Jean-Pierre (2003). Trees. Springer. pp. 56–57. ISBN 3-540-44237-5.