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  "body": "\n## Statement\n\nA useful consequence of the original Grushko theorem is the so-called **Grushko decomposition theorem.** It asserts that any nontrivial [[finitely generated group|082DWWXMRJ]] *G* can be decomposed as a free product\n\n*G* = *A*\u003csub\u003e1\u003c/sub\u003e∗*A*\u003csub\u003e2\u003c/sub\u003e∗...∗*A*\u003csub\u003e*r*\u003c/sub\u003e∗*F*\u003csub\u003e*s*\u003c/sub\u003e, where *s* ≥ 0, *r* ≥ 0,\n\nwhere each of the groups *A*\u003csub\u003e*i*\u003c/sub\u003e is nontrivial, freely indecomposable (that is, it cannot be decomposed as a free product) and not infinite cyclic, and where *F\u003csub\u003es\u003c/sub\u003e* is a [[free group|020Z5SEN27]] of rank *s*; moreover, for a given *G*, the groups *A*\u003csub\u003e1\u003c/sub\u003e, ..., *A*\u003csub\u003e*r*\u003c/sub\u003e are unique up to a permutation of their [[conjugacy classes|0DEBXAJ5SA]] 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.\n\nMore precisely, if *G* = *B*\u003csub\u003e1\u003c/sub\u003e∗...∗*B*\u003csub\u003e*k*\u003c/sub\u003e∗*F*\u003csub\u003e*t*\u003c/sub\u003e is another such decomposition then *k* = *r*, *s* = *t*, and there exists a permutation σ∈*S*\u003csub\u003e*r*\u003c/sub\u003e such that for each *i*=1,...,*r* the subgroups *A*\u003csub\u003e*i*\u003c/sub\u003e and *B*\u003csub\u003eσ(*i*)\u003c/sub\u003e are [[conjugate|0DEBXAJ5SA]] in *G*.\n\nThe 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).\n\nAlgorithmically 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.\n\nGrushko 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.\n",
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