Preservation of Trees by semidirect Products

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We show that the semidirect product of a group C by A*D B is isomorphic to the free product of A C and B C amalgamated at D C, where A, B and C are arbitrary groups. Moreover, we apply this theorem to prove that any group G that acts without inversion on a tree T that possesses a segment Γ for its quotient graph, such that, if the stabilizers of the vertex set {P,Q} and edge y of a lift of Γ in T are of the form Gₚ H, GQ H and Gy H, then G is isomorphic to the semidirect product of H by (Gₚ *Gy GQ). Using our results we conclude with a non-standard verification of the isomorphism between GL₂(Z) and the free product of the dihedral groups D₄ and D₆ amalgamated at their Klein-four group.