Character degrees, character codegrees and nilpotence class of p-groups

2021 ◽  
pp. 1-6
Author(s):  
Alexander Moretó
Keyword(s):  
Character Degrees ◽  
Nilpotence Class

scholarly journals 𝑝-groups with exactly four codegrees

2020 ◽  
Vol 23 (6) ◽  
pp. 1111-1122
Author(s):  
Sarah Croome ◽  
Mark L. Lewis
Keyword(s):  
Irreducible Character ◽  
Character Degree ◽  
Character Degrees ◽  
Additional Hypothesis ◽  
Nilpotence Class ◽  
Irreducible Character Degree

AbstractLet 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}}.

scholarly journals The Character Degrees and Nilpotence Class of a p-Group

2001 ◽  
Vol 238 (2) ◽  
pp. 827-842 ◽  
Author(s):  
I.M Isaacs ◽  
Alexander Moretó
Keyword(s):  
Character Degrees ◽  
Nilpotence Class

Character Codegrees of Maximal Class -groups

2019 ◽  
Vol 63 (2) ◽  
pp. 328-334
Author(s):  
Sarah Croome ◽  
Mark L. Lewis
Keyword(s):  
Irreducible Character ◽  
Character Degrees ◽  
Maximal Class ◽  
Class Groups ◽  
Nilpotence Class ◽  
Formula Group

AbstractLet $G$ 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.

scholarly journals Character degrees and nilpotence class in p-groups

1994 ◽  
Vol 57 (1) ◽  
pp. 76-80 ◽  
Author(s):  
Michael C. Slattery
Keyword(s):  
Irreducible Character ◽  
Character Degrees ◽  
Nilpotence Class ◽  
The Family ◽  
Non Linear

AbstractWork of Isaacs and Passman shows that for some sets X of integers, p-groups whose set of irreducible character degrees is precisely X have bounded nilpotence class, while for other choices of X, the nilpotence class is unbounded. This paper presents a theoren which shows some additional sets of character degrees which bound nilpotence class within the family of metabelian p-groups. In particular, it is shown that is the non-linear irreducible character degrees of G lie between pa and pb, where a ≤ b ≤ 2a − 2, then the nilpotence class of G is bounded by a function of p and b − a.

scholarly journals A new characterization of L2(p2)

2020 ◽  
Vol 18 (1) ◽  
pp. 907-915
Author(s):  
Zhongbi Wang ◽  
Chao Qin ◽  
Heng Lv ◽  
Yanxiong Yan ◽  
Guiyun Chen
Keyword(s):  
Positive Integer ◽  
Irreducible Character ◽  
Character Degrees ◽  
Prime Divisors

Abstract For a positive integer n and a prime p, let {n}_{p} denote the p-part of n. Let G be a group, \text{cd}(G) the set of all irreducible character degrees of G , \rho (G) the set of all prime divisors of integers in \text{cd}(G) , V(G)=\left\{{p}^{{e}_{p}(G)}|p\in \rho (G)\right\} , where {p}^{{e}_{p}(G)}=\hspace{.25em}\max \hspace{.25em}\{\chi {(1)}_{p}|\chi \in \text{Irr}(G)\}. In this article, it is proved that G\cong {L}_{2}({p}^{2}) if and only if |G|=|{L}_{2}({p}^{2})| and V(G)=V({L}_{2}({p}^{2})) .

Complex group algebras of the direct product of non-isomorphic Suzuki groups

2019 ◽  
Vol 19 (02) ◽  
pp. 2050036
Author(s):  
Morteza Baniasad Azad ◽  
Behrooz Khosravi
Keyword(s):  
Direct Product ◽  
Group Algebra ◽  
Irreducible Character ◽  
Prime Numbers ◽  
Product Formula ◽  
Character Degrees ◽  
Group Algebras ◽  
Complex Group ◽  
Complex Group Algebra ◽  
Suzuki Groups

In this paper, we prove that the direct product [Formula: see text], where [Formula: see text] are distinct numbers, is uniquely determined by its complex group algebra. Particularly, we show that the direct product [Formula: see text], where [Formula: see text]’s are distinct odd prime numbers, is uniquely determined by its order and three irreducible character degrees.

scholarly journals Taketa's theorem for relative character degrees

2013 ◽  
Vol 43 (5) ◽  
pp. 1451-1457
Author(s):  
Kamal Aziziheris
Keyword(s):  
Character Degrees

scholarly journals Actions and Invariant Character Degrees

1993 ◽  
Vol 160 (1) ◽  
pp. 172-178 ◽  
Author(s):  
G. Navarro
Keyword(s):  
Character Degrees
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