On the order of \(p\)-groups of abundance zero
Let \(G\) be a \(p\)-group and \(r(G)\) be the number of conjugacy classes in \(G\). P.~Hall proved that if \(|G|=p^{2n+e}\) with \(e\in\{0,1\}\) then \(r(G)=n(p^2-1)+p^e+l(p^2-1)(p-1)\) for some \(l\geq 0\). Then \(l=l(G)\) is the abundance of \(G\). It is proved that if \(p\geq 11\) is a prime, then there exist no \(p\)-groups of order \(p^{p+2}\) and abundance zero.
scientific article; zbMATH DE number 1214107