  There has been significant recent work which examines a situation where a
thin magnetic layer is `draped' over a core merging into a larger cluster; the
same process also appears to be at work at a bubble rising from the cluster
centre. Such a thin magnetic layer could thermally isolate the core from the
cluster medium, but only if the same shear process which generates the layer
does not later disrupt it. On the other hand, if the magnetized layer can
stabilize against the shear instabilities, then the magnetic layer can have the
additional dynamical effect of reducing the shear-driven mixing of the core's
material during the merger process. These arguments could equally well to
underdense cluster bubbles, which would be even more prone to disruption. While
it is well known that magnetic fields can suppress instabilities, it is less
clear that a thin layer can suppress instabilities on scales significantly
larger than its thickness. Here we consider the stability imparted by a thin
magnetized layer. Such a layer can have a significant stabilizing effect even
on modes with wavelengths much larger than the thickness of the layer l; to
stabilize modes ten times larger requires only that the Alfv\'en speed in the
magnetized layer is comparable to the relevant destabilizing velocity -- the
shear velocity in the case of pure Kelvin-Helmholtz like instability, or a
typical buoyancy velocity in the case of pure Rayleigh-Taylor. We confirm our
calculations with two-dimensional numerical experiments using the Athena code.
