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}} .

中文翻译：

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𝑝-具有正好四个 codegrees 的组

摘要 设 G 为 p 群，设 χ 为 G 的不可约特征。G : ker ( χ ) | / χ ( 1 ) {\lvert G:\operatorname{ker}(\chi)\rvert/\chi(1)} . Du 和 Lewis 已经证明，恰好具有三个密码的 p-group 具有至多 2 的幂零级。这里我们研究了恰好具有四个密码的 p-group。如果，除了正好有四个密码，G 有两个不可约字符度，G 有最大不可约字符度 p 2 {p^{2}} , | G : G ′ | = p 2 {\lvert G:G^{\prime}\rvert=p^{2}} ，或者 G 的 coclass 最多为 3，则 G 的幂等类最多为 4。 在 coclass 最多为 3 的情况下， G 的阶以 p 7 {p^{7}} 为界。有了一个额外的假设，我们可以将这个结果扩展到具有四个 codegrees 和 coclass 最多 6 个的 p-groups。在这种情况下，G 的阶数以 p 10 {p^{10}} 为界。