p-groups with a small number of character degrees and their normal subgroups

If G be a finite p-group and \chi is a non-linear irreducible character of G, then \chi(1)\leq |G/Z(G)|^{\frac{1}{2}}. In \cite{fernandez2001groups}, Fern\'{a}ndez-Alcober and Moret\'{o} obtained the relation between the character degree set of a finite p-group G and its normal subgroups depending on whether |G/Z(G)| is a square or not. In this paper we investigate the finite p-group G where for any normal subgroup N of G with G'\not \leq N either N\leq Z(G) or |NZ(G)/Z(G)|\leq p and obtain some alternate characterizations of such groups. We find that if G is a finite p-group with |G/Z(G)|=p^{2n+1} and G satisfies the condition that for any normal subgroup N of G either G'\not \leq N or N\leq Z(G), then cd(G)=\{1, p^{n}\}. We also find that if G is a finite p-group with nilpotency class not equal to 3 and |G/Z(G)|=p^{2n} and G satisfies the condition that for any normal subgroup N of G either G'\not \leq N or |NZ(G)/Z(G)|\leq p, then cd(G) \subseteq \{1, p^{n-1}, p^{n}\}.

p-groups with a small number of character degrees and their normal subgroups Keywords:

character degrees , p-groups , nilpotency class