Subgroup Isomorphism Problem for units of integral group rings

Abstract

Abstract The Subgroup Isomorphism Problem for Integral Group Rings asks for which finite groups U it is true that if U is isomorphic to a subgroup of V ⁢ ( ℤ ⁢ G ) ${\mathrm{V}(\mathbb{Z}G)}$ , the group of normalized units of the integral group ring of the finite group G, it must be isomorphic to a subgroup of G. The smallest groups known not to satisfy this property are the counterexamples to the Isomorphism Problem constructed by M. Hertweck. However, the only groups known to satisfy it are cyclic groups of prime power order and elementary-abelian p-groups of rank 2. We give a positive solution to the Subgroup Isomorphism Problem for C 4 × C 2 ${C_{4}\times C_{2}}$ . Moreover, we prove that if the Sylow 2-subgroup of G is a dihedral group, any 2-subgroup of V ⁢ ( ℤ ⁢ G ) ${\mathrm{V}(\mathbb{Z}G)}$ is isomorphic to a subgroup of G.