Class numbers of \(p\)-groups of a given order.
Let \(p\) be a prime number, \(G\) a finite \(p\)-group and \(k(G)\) the class number of \(G\). Let \(D(p^e)\) denote the number of different numbers \(k(G)\) that arise as \(G\) runs over all groups of order \(p^e\), \(e\in\mathbb{N}\). In this paper, the authors study the behavior of the number \(D(p^e)\) as \(p\) varies while the exponent \(e\) is held constant. It is known that \(D(p^e)\) is bounded for \(e\leq 6\). According to E. O'Brien, \(D(p^7)\) is bounded; he also believes that \(D(p^8)\) is also bounded. Examples presented in the paper show that \(D(p^9)\) approaches infinity as \(p\) approaches infinity. In the proof some deep results of algebraic geometry are used.
scientific article; zbMATH DE number 2104466