On Olsson’s conjecture for blocks with metacyclic defect groups of odd order

Let G be a finite group, and let B be a p-block of G with defect group D. Let k0(B) denote the number of ordinary irreducible characters of height 0 in B. In 1984 Olsson proposed a conjecture: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${k_{0}(B)\leqq |D:D'|}$$\end{document}. In this paper, we will verify Olsson’s conjecture in the case that D is metacyclic and p is odd.