𝑝-groups with exactly four codegrees

Abstract

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 ) {\lvert G:\operatorname{ker}(\chi)\rvert/\chi(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 p 2 {p^{2}} , | G : G ′ | = p 2 {\lvert G:G^{\prime}\rvert=p^{2}} , 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 p 7 {p^{7}} . 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 p 10 {p^{10}} .