Communicated by Michel BrouéKeywords: Irreducible character degree Group order upper bound Let d be the degree of an irreducible character of a finite group G.We can write |G| = d(d +e) for some non-negative integer e. In this document, we prove that if e > 1 then |G| < e 6 − e 4 . This improves an upper bound found by Isaacs of the form Be 6 , where B is an unknown universal constant. We also describe conditions sufficient to sharpen this bound to |G| e 4 − e 3 . In addition, we remove the appeal to the classification of simple groups, which is used in the original paper by Isaacs.

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