Subgroup Isomorphism Problem for Units of Integral Group Rings

TL;DR

This paper investigates the Subgroup Isomorphism Problem for units in integral group rings, proving it for specific groups and conditions, thereby advancing understanding of the structure of units related to finite groups.

Contribution

It proves the Subgroup Isomorphism Problem for the group C_4 x C_2 and for groups with dihedral Sylow 2-subgroups, extending known results in the field.

Findings

Proved the problem for C_4 x C_2.

Established that 2-subgroups are subgroups of G when Sylow 2-subgroup is dihedral.

Extended the class of groups satisfying the Subgroup Isomorphism Problem.

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(ZG), 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 prove the Subgroup Isomorphism Problem for C_4 x C_2. Moreover we prove that if the Sylow 2-subgroup of G is a dihedral group, any 2-subgroup of V(ZG) is isomorphic to a subgroup of G.