  Since it has been pointed out that physics beyond the Standard Model may be
constrained by gravitational waves from cosmic strings, it has been more
important to clarify in what cases cosmic strings are formed. We study the
stability of the embedded string which is formed when $SU(N)\times U(1)_X$
gauge symmetries are broken to $SU(N-1)\times U(1)_Q$, and find that the
stability condition can be determined by two mass ratios of the Higgs and
massive gauge bosons, and does not explicitly depend on $N$. We also show that
the result can be extended in supersymmetric models. In addition, we apply
these results to several models and discuss the important feature of the Higgs
to produce the embedded string. Although we find it difficult to be satisfied
in normal realistic GUT models, it is possible if $SU(N)$ and $U(1)_X$ have
different origins.
