  We systematically investigate the robustness of symmetry protected
topological (SPT) order in open quantum systems by studying the evolution of
string order parameters and other probes under noisy channels. We find that
one-dimensional SPT order is robust against noisy couplings to the environment
that satisfy a strong symmetry condition, while it is destabilized by noise
that satisfies only a weak symmetry condition, which generalizes the notion of
symmetry for closed systems. We also discuss "transmutation" of SPT phases into
other SPT phases of equal or lesser complexity, under noisy channels that
satisfy twisted versions of the strong symmetry condition.
