[Paper Review] Character codegrees of maximal class p-groups

Sarah Croome, Mark L. Lewis|arXiv (Cornell University)|Sep 20, 2018

Finite Group Theory Research4 references4 citations

TL;DR

This paper investigates the codegrees of irreducible characters in maximal class $p$-groups, proving that under specific conditions—such as being normally monomial or having few character degrees—these codegrees are consecutive powers of $p$. The key result establishes that if a $p$-group has all powers of $p$ from $p^0$ to $p^{n-1}$ as codegrees, then it must either be abelian-by-cyclic of class 2 or have maximal class with exactly two character degrees.

ABSTRACT

Let $G$ be a $p$-group and let $χ$ be an irreducible character of $G$. The codegree of $χ$ is given by $|G: ext{ker}(χ)|/χ(1)$. 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.

Motivation & Objective

• To determine the structure of maximal class $p$-groups whose character codegrees form a complete set of consecutive $p$-powers.
• To investigate whether the codegrees of maximal class $p$-groups are always consecutive powers of $p$, especially in relation to nilpotence class and character degree sets.
• To explore the role of normally monomial and metabelian structures in constraining codegree sets.
• To establish necessary and sufficient conditions under which the full range of $p$-powers up to $p^{n-1}$ appears as codegrees in a $p$-group of order $p^n$.

Proposed method

• Define the codegree of an irreducible character $\chi$ as $\text{cod}(\chi) = |G:\ker(\chi)| / \chi(1)$, and study $\text{cod}(G)$, the set of all such codegrees.
• Use induction on the quotient group $G/Z$ for normally monomial $p$-groups, leveraging the fact that quotients of normally monomial groups are also normally monomial.
• Apply Itô’s Theorem and results from character induction to analyze faithful irreducible characters and their degrees in $p$-groups.
• Use the upper and lower central series to analyze nilpotence class and relate it to character degree and codegree structure.
• Leverage known results on $p$-groups with few character degrees, such as $\text{cd}(G) = \{1,p\}$, to classify groups with maximal codegree sets.
• Prove that $p^4 \in \text{cod}(G)$ for maximal class $p$-groups of order at least $p^6$, using quotient group analysis and character degree bounds.

Experimental results

Research questions

• RQ1 Under what conditions do maximal class $p$-groups have codegrees that are all consecutive powers of $p$?
• RQ2 What is the relationship between the number of character degrees and the structure of codegrees in maximal class $p$-groups?
• RQ3 Can the codegrees of a maximal class $p$-group fail to include $p^k$ for some $k$ in the expected range, and if so, under what conditions?
• RQ4 How does the normally monomial property constrain the codegree set of a maximal class $p$-group?
• RQ5 Is there a structural characterization of $p$-groups whose codegrees span all $p$-powers from $p^0$ to $p^{n-1}$?

Key findings

• A $p$-group $G$ of order $p^n$ has $\text{cod}(G) = \{p^i \mid 0 \leq i \leq n-1\}$ if and only if it is isomorphic to $\mathbb{Z}_{p^{n-1}} \times \mathbb{Z}_p$, or a class 2 group with $n \geq 4$, or a maximal class group with exactly two character degrees.
• For a maximal class $p$-group with $\text{cd}(G) = \{1, p, p^b\}$, the codegrees are $\{p^i \mid 0 \leq i \leq c\}$ for some $c$ satisfying $n - b \leq c \leq n - 2$.
• In metabelian maximal class $p$-groups, the codegrees are consecutive powers of $p$, with the largest codegree being $p^{n-1}$ or $p^{n-2}$.
• For normally monomial maximal class $p$-groups of order $p^n$, the codegrees are $\{p^i \mid 0 \leq i \leq c\}$ with $c \geq n - b$, where $b = \log_p(b(G))$ and $b(G)$ is the largest character degree.
• If $|G| \geq p^6$, then $p^4 \in \text{cod}(G)$ for any maximal class $p$-group $G$, and this bound is sharp.
• There exist maximal class $p$-groups of order $p^5$ (e.g., SmallGroup(3^5,i) for i=28,29,30) for which $p^4 \notin \text{cod}(G)$, showing that the $p^4$ condition is not guaranteed below order $p^6$.