FINITE p-GROUPS WITH FEW NON-LINEAR IRREDUCIBLE CHARACTER KERNELS

We classify finite p-groups with at most three non-linear irreducible character kernels. This solves the first half of a question posed by Berkovich [1, Research Problem 23]. The second part of this problem involves the quasikernels and is still open. Throughout the paper, G is a finite non-abelian p-group for a fixed prime p. Denote by Kern(G) the set of non-linear irreducible character kernels of G. If |Kern(G)| = 1, then |G | = p and Z(G) is cyclic and vice versa (see Lemma 2.1 and Lemma 2.2 below). Also, the main theorem of [8] implies that if |Kern(G)| 1, then G is of maximal class if and only if Kern(G) is a chain with respect to inclusion. Our main theorem is the following.