The Subgroup Isomorphism Problem for Integral Group Rings asks for whichfinite groups U it is true that if U is isomorphic to a subgroup of V(ZG), thegroup of normalized units of the integral group ring of the finite group G, itmust be isomorphic to a subgroup of G. The smallest groups known not to satisfythis property are the counterexamples to the Isomorphism Problem constructed byM. Hertweck. However the only groups known to satisfy it are cyclic groups ofprime power order and elementary-abelian p-groups of rank 2. We prove theSubgroup Isomorphism Problem for C_4 x C_2. Moreover we prove that if the Sylow2-subgroup of G is a dihedral group, any 2-subgroup of V(ZG) is isomorphic to asubgroup of G.
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机译：整数环的子群同构问题询问哪个有限群U，如果U与V（ZG）的一个子群同构，则有限群G的整数环的归一化单元群必须同构。已知不满足此特性的最小组是M构造的同构问题的反例。赫特维克。但是，已知满足它的唯一组是素数次幂的循环组和等级2的基本阿贝尔p-组。我们证明了C_4 x C_2的子组同构问题。此外，我们证明如果G的Sylow2-子群是二面体基团，则V（ZG）的任何2-子群与G的子群同构。
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