The t-core of an s-core

We consider the t-core of an s-core partition, when s and t are coprime positive integers. Olsson has shown that the t-core of an s-core is again an s-core, and we examine certain actions of the affine symmetric group on s-cores which preserve the t-core of an s-core. Along the way, we give a new proof of Olssonʼs result. We also give a new proof of a result of Vandehey, showing that there is a simultaneous s- and t-core which contains all others.