## A result about Characters of F-algebra groups

Let G be an F-algebra group(G=1+J , where J is the jacobson radical of a finite dimensional F-algebra ,where F is a field of prime characteristic) In a paper of Isaacs ("Characters of groups associated with finite algebras" from 1995) there is a claim of Gutkin with a wrong proof.It says: Let x be an irreducible character of G ,then $x=a^G$ ,where a is a linear character of some subgroup H<=G of the form : H=1+U where U is multiplicativly closed F-subspace of J. This result would be a generalisation of the result of Isaacs paper,but its unclear to me if it has been proven until today.Does someone know if the result is true and can recommand me a paper/link for more information?

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I think the result you are referring to is over a finite field $F$ (Gutkin's claim was also stated over local fields, however).