Groups of prime power order with many conjugacy classes
Concerning the number \(k(G)\) of conjugacy classes of a finite \(p\)-group \(G\) the authors investigate on the one hand the structure of groups with relatively large \(k(G)\), on the other hand they give estimates of values, \(k(G)\) cannot achieve. With respect to the latter question they derive for example the following result: There exists no \(p\)-group \(G\) (\(p>2\)) of order \(p^n\) such that \(3p^{n-3}\leq k(G)\leq p^{n-2}+2p-3\). Regarding the former problem Theorem 1 states, that \(k(G)\geq 3| G|/p^3\) implies (i) \(| G_2|\leq p^2\) or (ii) \(| G:Z(G)|\leq p^3\), or (iii) \(G\) has an abelian maximal subgroup. A similar result is proved if \(k(G)\geq 5| G|/p^4\). Corollaries relate among others to \(p\)-groups of breadth 2 or 3. Using other methods and thereby eliminating some excluded cases the results on the breadth of \(G\) have been proved just recently by \textit{G. Parmeggiani} and \textit{B. Stellmacher} [in J. Algebra 213, No. 1, 52-68 (1999; see the following review Zbl 0921.20018)].
scientific article; zbMATH DE number 1203887