The twisted group ring isomorphism problem over fields
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THE TWISTED GROUP RING ISOMORPHISM PROBLEM OVER FIELDS
BY
Leo Margolis∗ Department of Mathematics, Free University of Brussels, 1050 Brussels, Belgium e-mail: [email protected]
AND
Ofir Schnabel Department of Mathematics, ORT Braude College, 2161002 Karmiel, Israel e-mail: [email protected]
ABSTRACT
Similarly to how the classical group ring isomorphism problem asks, for a commutative ring R, which information about a finite group G is encoded in the group ring RG, the twisted group ring isomorphism problem asks which information about G is encoded in all the twisted group rings of G over R. We investigate this problem over fields. We start with abelian groups and show how the results depend on the characteristic of R. In order to deal with non-abelian groups we construct a generalization of a Schur cover which exists also when R is not an algebraically closed field, but still linearizes all projective representations of a group. We then show that groups from the celebrated example of Everett Dade which have isomorphic group algebras over any field can be distinguished by their twisted group algebras over finite fields.
∗ The first author is a postdoctoral researcher of the Research Foundation Flan-
ders (FWO—Vlaanderen). We are grateful for the Technion—Israel Institute of Technology, for supporting the first author’s visit to Haifa Received February 12, 2019 and in revised form June 6, 2019
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L. MARGOLIS AND O. SCHNABEL
Isr. J. Math.
1. Introduction In [MS18] we proposed a twisted version of the celebrated group ring isomorphism problem (GRIP), namely “the twisted group ring isomorphism problem”(TGRIP). Recall that for a finite group G and a commutative ring R, the group ring isomorphism problem asks whether the ring structure of RG determines G up to isomorphism. In other words, does the existence of a ring isomorphism RG ∼ = RH imply the existence of a group isomorphism G ∼ = H, for given groups G and H? Roughly speaking the twisted group ring isomorphism problem asks if for a group G and a commutative ring R, the ring structure of all the twisted group rings of G over R determines the group G. The role twisted group rings of G over R play for the projective representation theory is in many ways the same played by the group ring RG for the representation theory of G over R, as it was shown in the ground laying work of I. Schur [Sch07]. In this sense the (TGRIP) can also be understood as a question on how strongly the projective representation theory of a group influences its structure. For results on the classical (GRIP) see [RS87, Seh93, Her01] for the case R = Z. Also questions on character degrees, as addressed, e.g., in [Isa76, Nav18], can be viewed as results for the case R = C. We denote by R∗ the unit group in a ring R. For a 2-cocycle α ∈ Z 2 (G, R∗ ) the twisted group ring Rα G of G over R with respect to α is the free R-module with basis {ug }g∈G where the multiplication on the basis is defined via ug uh = α(g, h)ugh
for all g, h ∈ G
and any ug commutes with the elements of R. Notice that if we
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