Conjugacy class lengths of metanilpotent groups

We show that in a finite group $G$ which is $p$ -nilpotent for at most one prime dividing its order, there exists an element whose conjugacy class length is divisible by more than half of the primes dividing $∣G / Z (G)∣$ .

Finite groups with primitive Sylow normalizers

A. D&amp;#039;Aniello, C. De Vivo, G. Giordano (2002)

Bollettino dell'Unione Matematica Italiana

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We prove that are primitive the finite groups whose normalizers of the Sylow subgroups are primitive. We classify the groups of such class, denoted by $N P$ , and we study the Schunck classes whose boundary is contained in $N P$ . We give also necessary and sufficient conditions in order that the projectors be subnormally embedded.

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