𝑝-groups with exactly four codegrees

Let G be a p-group, and let χ be an irreducible character of G. The codegree of χ is given by {\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}}, {\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}}. 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}}.

Character Codegrees of Maximal Class -groups

Let $G$ be a $p$-group and let $\unicode[STIX]{x1D712}$ be an irreducible character of $G$. The codegree of $\unicode[STIX]{x1D712}$ is given by $|G:\,\text{ker}(\unicode[STIX]{x1D712})|/\unicode[STIX]{x1D712}(1)$. If $G$ is a maximal class $p$-group that is normally monomial or has at most three character degrees, then the codegrees of $G$ are consecutive powers of $p$. If $|G|=p^{n}$ and $G$ has consecutive $p$-power codegrees up to $p^{n-1}$, then the nilpotence class of $G$ is at most 2 or $G$ has maximal class.

Centralizers of Camina p-groups of nilpotence class 3

LetGbe a Camina\hskip-0.853583pt{p}-group of nilpotence class 3. We prove that if{G^{\prime}\hskip-0.853583pt<\hskip-0.853583ptC_{G}(G^{\prime})}, then{|G_{3}|\leq|G^{\prime}:G_{3}|^{1/2}}. We also prove that if{G/G_{3}}has only one or two abelian subgroups of order{|G:G^{\prime}|}, then{G^{\prime}<C_{G}(G^{\prime})}. If{G/G_{3}}has{p^{a}+1}abelian subgroups of order{|G:G^{\prime}|}, then either{G^{\prime}<C_{G}(G^{\prime})}or{|Z(G)|\leq p^{2a}}.

Codegrees and nilpotence class of p-groups

If χ is an irreducible character of a finite group

A pair of characteristic subgroups for pushing-up. II

Many problems about local analysis in a finite group G reduce to a special case in which G has a large normal p-subgroup satisfying several restrictions. In 1983, R. Niles and G. Glauberman showed that every finite p-group S of nilpotence class at least 4 must have two characteristic subgroups S1 and S2 such that, whenever S is a Sylow p-subgroup of a group G as above, S1 or S2 is normal in G. In this paper, we prove a similar theorem with a more explicit choice of S1 and S2.