In the mathematical subject of group theory, the Grushko theorem or the Grushko–Neumann theorem is a theorem stating that the rank (that is, the smallest cardinality of a generating set) of a free product of two groups is equal to the sum of the ranks of the two free factors. The theorem was first obtained in a 1940 article of Grushko and then, independently, in a 1943 article of Neumann.
Statement of the theorem Let A and B be finitely generated groups and let A∗B be the free product of A and B. Then
rank(A∗B) = rank(A) + rank(B). It is obvious that rank(A∗B) ≤ rank(A) + rank(B) since if X is a finite generating set of A and Y is a finite generating set of B then X∪Y is a generating set for A∗B and that |X ∪ Y| ≤ |X| + |Y|. The opposite inequality, rank(A∗B) ≥ rank(A) + rank(B), requires proof. Grushko, but not Neumann, proved a more precise version of Grushko's theorem in terms of Nielsen equivalence. It states that if M = (g1, g2, ..., gn) is an n-tuple of elements of G = A∗B such that M generates G, <g1, g2, ..., gn> = G, then M is Nielsen equivalent in G to an n-tuple of the form
M' = (a1, ..., ak, b1, ..., bn−k) where {a1, ..., ak}⊆A is a generating set for A and where {b1, ..., bn−k}⊆B is a generating set for B. In particular, rank(A) ≤ k, rank(B) ≤ n − k and rank(A) + rank(B) ≤ k + (n − k) = n. If one takes M to be the minimal generating tuple for G, that is, with n = rank(G), this implies that rank(A) + rank(B) ≤ rank(G). Since the opposite inequality, rank(G) ≤ rank(A) + rank(B), is obvious, it follows that rank(G)=rank(A) + rank(B), as required.
History and generalizations After the original proofs of Grushko (1940) and Neumann(1943), there were many subsequent alternative proofs, simplifications and generalizations of Grushko's theorem. A close version of Grushko's original proof is given in the 1955 book of Kurosh. Like the original proofs, Lyndon's proof (1965) relied on length-functions considerations but with substantial simplifications. A 1965 paper of Stallings gave a greatly simplified topological proof of Grushko's theorem. A 1970 paper of Zieschang gave a Nielsen equivalence version of Grushko's theorem (stated above) and provided some generalizations of Grushko's theorem for amalgamated free products. Scott (1974) gave another topological proof of Grushko's theorem, inspired by the methods of 3-manifold topology Imrich (1984) gave a version of Grushko's theorem for free products with infinitely many factors. A 1976 paper of Chiswell gave a relatively straightforward proof of Grushko's theorem, modelled on Stallings' 1965 proof, that used the techniques of Bass–Serre theory. The argument directly inspired the machinery of foldings for group actions on trees and for graphs of groups and Dicks' even more straightforward proof of Grushko's theorem (see, for example, ). Grushko's theorem is, in a sense, a starting point in Dunwoody's theory of accessibility for finitely generated and finitely presented groups. Since the ranks of the free factors are smaller than the rank of a free product, Grushko's theorem implies that the process of iterated splitting of a finitely generated group G as a free product must terminate in a finite number of steps (more precisely, in at most rank(G) steps). There is a natural similar question for iterating splittings of finitely generated groups over finite subgroups. Dunwoody proved that such a process must always terminate if a group G is finitely presented but may go on forever if G is finitely generated but not finitely presented. An algebraic proof of a substantial generalization of Grushko's theorem using the machinery of groupoids was given by Higgins (1966). Higgins' theorem starts with groups G and B with free decompositions G = ∗i Gi, B = ∗i Bi and f : G → B a morphism such that f(Gi) = Bi for all i. Let H be a subgroup of G such that f(H) = B. Then H has a decomposition H = ∗i Hi such that f(Hi) = Bi for all i. Full details of the proof and applications may also be found in .
Grushko decomposition theorem A useful consequence of the original Grushko theorem is the so-called Grushko decomposition theorem. It asserts that any nontrivial finitely generated group G can be decomposed as a free product
G = A1∗A2∗...∗Ar∗Fs, where s ≥ 0, r ≥ 0, where each of the groups Ai is nontrivial, freely indecomposable (that is, it cannot be decomposed as a free product) and not infinite cyclic, and where Fs is a free group of rank s; moreover, for a given G, the groups A1, ..., Ar are unique up to a permutation of their conjugacy classes in G (and, in particular, the sequence of isomorphism types of these groups is unique up to a permutation) and the numbers s and r are unique as well. More precisely, if G = B1∗...∗Bk∗Ft is another such decomposition then k = r, s = t, and there exists a permutation σ∈Sr such that for each i=1,...,r the subgroups Ai and Bσ(i) are conjugate in G. The existence of the above decomposition, called the Grushko decomposition of G, is an immediate corollary of the original Grushko theorem, while the uniqueness statement requires additional arguments (see, for example). Algorithmically computing the Grushko decomposition for specific classes of groups is a difficult problem which primarily requires being able to determine if a given group is freely decomposable. Positive results are available for some classes of groups such as torsion-free word-hyperbolic groups, certain classes of relatively hyperbolic groups, fundamental groups of finite graphs of finitely generated free groups and others. Grushko decomposition theorem is a group-theoretic analog of the Kneser prime decomposition theorem for 3-manifolds which says that a closed 3-manifold can be uniquely decomposed as a connected sum of irreducible 3-manifolds.
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