Representation dimensions linked by Frobenius bimodules with applications to group algebras

We establish relations between representation dimensions of two algebras connected by a Frobenius bimodule or extension. Consequently, upper bounds and equality formulas for representation dimensions of group algebras, symmetric separably equivalent algebras and crossed products are obtained. Particularly, for any subgroup H of a finite group G, if [G:H] is invertible in a field, then the representation dimensions of the group algebras of G and H over the field are the same.