On Camina groups of prime power order
The authors define a `Camina group' as one in which \(|C_G(g)|=|C_{G/G'}(gG')|\) for all \(g\) in \(G\setminus G'\). The main result is that if \(G\) is a finite Camina group then \(G\) has nilpotency class at most 3. The proof of the result is contained in Section 3 which is a very neat argument using earlier work of \textit{L. Verardi} [Ann. Mat. Pura Appl., IV. Ser. 148, 131-171 (1987; Zbl 0648.20032)] and \textit{A. Mann} and \textit{C. M. Scoppola} [Arch. Math. 56, No. 4, 320-332 (1991; Zbl 0788.20010)] to get bounds on the orders of various factors of \(G\). They then use a Lie algebra argument to complete the proof. Section 4 is devoted to the construction of class 3 Camina groups. In this they return to the idea in Section 3 to construct their examples and show how the existence of certain skew-symmetric matrices, where entries are homogeneous polynomials, give rise to the examples they are looking for.
scientific article; zbMATH DE number 883786