Abstract
We prove that if two finite groups G1 and G2 admit an isomorphism between their lattices of subgroups which preserves subgroup rings over a commutative ring K then the partial group rings Kpar G1 and Kpar G2 are isomorphic. If |G| is odd, then we find a series of natural invariants of ℂpar G and use them to show that ℂpar G determines the commutativity of G. In the case of odd |G|, the least counterexample to the isomorphism problem for partial group algebras over ℂ is pointed out. Moreover, a counterexample to the isomorphism problem over ℚ is also given.
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