Finite $p$-groups with small number of degrees of irreducible character

Abstract

Let $G$ be a finite $p$-group with ${\rm{cd}}(G)=\{1,p^m\}$. In  this paper, we prove that the order of  the commutator subgroup satisfies $|G'|\ge p^{(n-2)m+1}$ if $m\ge 2$ and the nilpotent class $c(G)=n$. We also classify the finite $p$-group $G$ with $G'$ cyclic and the nilpotent class $c(G)=2$.

Keywords

• finite p-group
• character
• degree of character
• nilpotent class

Supporting Institution

National Natural Science Foundation of China, Fundamental Research Funds for the Central Universities

Project Number

No.11971391, Grant No. SWU-KT25017 and Grant No.SWU-XDJH202305

Ethical Statement

Not applicable. This research involves no human participants or animals.

References

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