𝑝-groups with exactly four codegrees

Sarah Croome 1  and Mark L. Lewis 2
  • 1 Department of Mathematical Sciences, Kent State University, OH 44242, Kent, USA
  • 2 Department of Mathematical Sciences, Kent State University, OH 44242, Kent, USA
Sarah Croome and Mark L. Lewis

Abstract

Let G be a p-group, and let χ be an irreducible character of G. The codegree of χ is given by |G:ker(χ)|/χ(1). Du and Lewis have shown that a p-group with exactly three codegrees has nilpotence class at most 2. Here we investigate p-groups with exactly four codegrees. If, in addition to having exactly four codegrees, G has two irreducible character degrees, G has largest irreducible character degree p2, |G:G|=p2, or G has coclass at most 3, then G has nilpotence class at most 4. In the case of coclass at most 3, the order of G is bounded by p7. With an additional hypothesis, we can extend this result to p-groups with four codegrees and coclass at most 6. In this case, the order of G is bounded by p10.

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The Journal of Group Theory is devoted to the publication of original research articles in all aspects of group theory. Articles concerning applications of group theory and articles from research areas which have a significant impact on group theory will also be considered.

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