I will discuss the problem of ground state factorization in spin chains. After a general introduction about the topic, I will describe in detail the case of the XY chain in a transverse field. It will be shown that the factorized ground state is an accidental degeneracy of the Hamiltonian. A finite-size analysis allows one to establish the cause of the existence of different regimes in the symmetry-broken phase and to emphasize the role of the factorizing field in the process. In the second part of the seminar, I will extend the study to dimerized chains, where, together with the usual ferromagnetic and antiferromagnetic regimes, a third case emerges, consisting of an antiferromagnetic Neél-type ground state whose unitary cell is represented by a pairs of spins. As in the regular XY model, the factorizing field represents a border line between region exhibiting different symmetry-breaking mechanisms