# (Non-)Morita invariance of the isomorphism type of a defect group

The modular isomorphism problem is a special case of the question of whether the isomorphism type of a defect group is a Morita invariant, and in [GMdelR21] the modular isomorphism problem was shown to have a negative answer . Pairs of nonisomorphic 2-groups are given whose group algebra over a field of two elements are isomorphic, hence their group algebras over any field of characteristic 2 are isomorphic. Since the group algebras of p-groups are basic, it follows directly that the group algebras a Morita equivalent.

The smallest pair of examples is SmallGroup(512,453) and SmallGroup(512,456). These groups have cyclic derived subgroup of order four and nilpotency class three. They have presentations \[ \langle x,y,z \mid z=[y,x],x^{16}=y^8=z^4=1, z^x=z^y=z^{-1} \rangle \] and \[ \langle x,y,z \mid z=[y,x],x^{16}=y^8=z^4=1, z^x=z^{-1}, z^y=z \rangle \] respectively.

Note that the group algebras over a complete local principal ideal domain with residue field of characteristic two cannot be isomorphic. Hence they also give examples of blocks that are Morita equivalent of a field of characteristic p but not over a complete local principal ideal domain for which it is a residue field.