Annals of Mathematics · 2001 · 150 citations · 6 references
Natural SciencesMathematicsAlgebraGroup WorkRepresentation TheoryAbstract AlgebraGroup StructureGroup TheoryFinite Basis Problem
The isomorphism problem for integral group rings asks whether a finite group X is determined by its integral group ring ZX, where a group basis of ZX is a subgroup Y of units of augmentation 1 satisfying ZX = ZY and IXI = YI. The construction uses a subgroup G of X with a non‑inner automorphism that becomes inner in ZG. The authors provide a finite group X whose integral group ring ZX has a group basis not isomorphic to X, giving a counterexample to the isomorphism problem. It was.
Let X be a finite group, and denote its integral group ring by ZX. A group basis of ZX is a subgroup Y of the group of units of ZX of augmentation 1 such that ZX = ZY and IXI = YI. An example of a finite group X is given such that ZX has a group basis which is not isomorphic to X. A main ingredient is the existence of a subgroup G of X which possesses a non-inner automorphism which becomes inner in the integral group ring ZG. The question whether a finite group X is determined by its integral group ring ZX is known as the 'isomorphism problem for integral group rings'. It was
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Alfred Weiss · Annals of Mathematics · 1988 · 114 citations
Isomorphisms of p-adic Group Rings
Klaus W. Roggenkamp, Leonard L. Scott · Annals of Mathematics · 1987 · 104 citations